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Random Number Generation and Sampling Methods

, 5 Oct 2018
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Discusses many ways applications can do random number generation and sampling from an underlying random number generator (RNG) and includes pseudocode for many of them.

Introduction

This page discusses many ways applications can do random number generation and sampling from an underlying random number generator (RNG), often with pseudocode. Those methods include—

Sample Python code that implements many of the methods in this document is available.

All the random number methods presented on this page are ultimately based on an underlying RNG; however, the methods make no assumptions on that RNG's implementation (e.g., whether that RNG is a deterministic RNG or some other kind) or on that RNG's statistical quality or predictability.

In general, this document does not cover:

  • How to choose an underlying RNG for a particular application, including in terms of security, performance, and quality. I have written more on RNG recommendations in another document.
  • Techniques that are specific to certain kinds of RNGs or application programming interfaces.
  • Seemingly random numbers that are specifically generated using hash functions, including pseudorandom functions (as opposed to RNGs). But if such a number is used to initialize a deterministic RNG (that is, to serve as its "seed"), then that RNG is generally within the scope of this document.

About This Document

This is an open-source document; for an updated version, see the source code or its rendering on GitHub. You can send comments on this document either on CodeProject or on the GitHub issues page.

Comments on any aspect of this document are welcome, but especially comments on the following:

  • Corrections to any method given on this page.
  • Requests to provide an implementation of any method given here in other programming languages, in addition to Python.
  • If there is enough interest by readers, I may discuss approaches to generate random mazes, graphs, matrices, or paths.
  • Suggestions to trim the size of this document, such as by limiting it to the most common and most useful methods for generating random numbers.

Contents

Notation and Definitions

  • The pseudocode conventions apply to this document.
  • Intervals. The following notation is used for intervals:
    • [a, b) means "a or greater, but less than b".
    • (a, b) means "greater than a, but less than b".
    • (a, b] means "greater than a and less than or equal to b".
    • [a, b] means "a or greater and b or less".
  • Norm. The norm of one or more real numbers is the square root of the sum of squares of those numbers, that is, sqrt(num1 * num1 + num2 * num2 + ... + numN * numN).
  • Random number generator (RNG). Software and/or hardware that seeks to generate independent numbers that seem to occur by chance and that are approximately uniformly distributed(1).

Uniform Random Numbers

This section describes how an underlying RNG can be used to generate uniformly-distributed random numbers. Here is an overview of the methods described in this section.

  • Random Integers: RNDINT, RNDINTEXC, RNDINTRANGE, RNDINTRANGEEXC.
  • Random Numbers in 0-1 Bounded Interval: RNDU01, RNDU01ZeroExc, RNDU01OneExc, RNDU01ZeroOneExc.
  • Other Random Numbers: RNDNUMRANGE, RNDNUMRANGEEXC.

One method, RNDINT, described next, can serve as the basis for the remaining methods.

RNDINT: Random Integers in [0, N]

In this document, RNDINT(maxInclusive) is the core method for generating independent uniform random integers from an underlying RNG, which is called RNG() in this section. The random integer is in the interval [0, maxInclusive]. If RNG() outputs integers in the interval [0, positive MODULUS) (examples of MODULUS include 1,000,000 and 6), then RNDINT(maxInclusive) can be implemented as in the pseudocode below.(2) (RNGs that output numbers in the interval [0, 1), such as Wichmann–Hill and dSFMT, are also seen in practice, but building RNDINT for those RNGs is more complicated unless the set of numbers they could return, as opposed to their probability, is evenly distributed.)

 

METHOD RndIntHelperNonPowerOfTwo(maxInclusive)
    cx = floor(maxInclusive / MODULUS) + 1
    while true
       ret = cx * RNG()
       // NOTE: If this method is implemented using a fixed-
       // precision type, the addition operation below should
       // check for overflow and should reject the number
       // if overflow would result.
       ret = ret + RNDINT(cx - 1)
       if ret <= maxInclusive: return ret
    end
END METHOD

METHOD RndIntHelperPowerOfTwo(maxInclusive)
        // NOTE: Finds the number of bits minus 1 needed
        // to represent MODULUS (in other words, the number
        // of random bits returned by RNG() ). This will
        // be a constant here, though.
        modBits = ln(MODULUS)/ln(2)
        // Calculate the bit count of maxInclusive
        bitCount = 0
        tempnumber = maxInclusive
        while tempnumber > 0
               // NOTE: If the programming language implements
               // division with two integers by truncating to an
               // integer, the division can be used as is without
               // using a "floor" function.
               tempnumber = floor(tempnumber / 2)
               bitCount = bitCount + 1
        end
        while true
               // Build a number with `bitCount` bits
                tempnumber = 0
                while bitCount > 0
                     wordBits = modBits
                     rngNumber = RNG()
                     if wordBits > bitCount
                        wordBits = bitCount
                        // Truncate number to 'wordBits' bits
                        // NOTE: If the programming language supports a bitwise
                        // AND operator, the mod operation can be implemented
                        // as "rndNumber AND ((1 << wordBits) - 1)"
                        rngNumber = rem(rngNumber, (1 << wordBits))
                     end
                     tempnumber = tempnumber << wordBits
                     // NOTE: In programming languages that
                     // support the OR operator between two
                     // integers, that operator can replace the
                     // plus operator below.
                     tempnumber = tempnumber + rngNumber
                     bitCount = bitCount - wordBits
                end
                // Accept the number if allowed
                if tempnumber <= maxInclusive: return tempnumber
         end
END METHOD

METHOD RNDINT(maxInclusive)
  // maxInclusive must be 0 or greater
  if maxInclusive < 0: return error
  if maxInclusive == 0: return 0
  // N equals modulus
  if maxInclusive == MODULUS - 1: return RNG()
  // NOTE: Finds the number of bits minus 1 needed
  // to represent MODULUS (if it's a power of 2).
  // This will be a constant here, though.
  modBits=ln(MODULUS)/ln(2)
  // NOTE: The following condition checks if MODULUS
  // is a power of 2.  This will be a constant here, though.
  isPowerOfTwo=floor(modBits) == modBits
  // Special cases if MODULUS is a power of 2
  if isPowerOfTwo
       if maxInclusive == 1: return rem(RNG(), 2)
       if maxInclusive == 3 and modBits >= 2: return rem(RNG(), 4)
       if maxInclusive == 255 and modBits >= 8: return rem(RNG(), 256)
       if maxInclusive == 65535 and modBits >=16: return rem(RNG(), 65535)
   end
  if maxInclusive > MODULUS - 1:
     if isPowerOfTwo
       return RndIntHelperPowerOfTwo(maxInclusive)
     else
       return RndIntHelperNonPowerOfTwo(maxInclusive)
     end
  else
    // NOTE: If the programming language implements
    // division with two integers by truncating to an
    // integer, the division can be used as is without
    // using a "floor" function.
          nPlusOne = maxInclusive + 1
          maxexc = floor((MODULUS - 1) / nPlusOne) * nPlusOne
          while true
                    ret = RNG()
                    if ret < nPlusOne: return ret
                    if ret < maxexc: return rem(ret, nPlusOne)
          end
  end
END METHOD

Notes:

  • To generate a random number that's either -1 or 1, the following idiom can be used: (RNDINT(1) * 2 - 1).
  • To generate a random integer that's divisible by a positive integer (DIV), generate the integer with any method (such as RNDINT), let X be that integer, then generate X - rem(X, DIV) if X >= 0, or X - (DIV - rem(abs(X), DIV)) otherwise. (Depending on the method, the resulting integer may be out of range, in which case this procedure is to be repeated.)
  • A random 2-dimensional point on an NxM grid can be expressed as a single integer as follows:
    • To generate a random NxM point P, generate P = RNDINT(N * M - 1) (P is thus in the interval [0, N * M)).
    • To convert a point P to its 2D coordinates, generate [rem(P, N), floor(P / N)]. (Each coordinate starts at 0.)
    • To convert 2D coordinates coord to an NxM point, generate P = coord[1] * N + coord[0].

RNDINTRANGE: Random Integers in [N, M]

The naïve way of generating a random integer in the interval [minInclusive, maxInclusive], shown below, works well for unsigned integers and arbitrary-precision integers.

METHOD RNDINTRANGE(minInclusive, maxInclusive)
  // minInclusive must not be greater than maxInclusive
  if minInclusive > maxInclusive: return error
  return minInclusive + RNDINT(maxInclusive - minInclusive)
END METHOD

The naïve approach won't work as well, though, for signed integer formats if the difference between maxInclusive and minInclusive exceeds the highest possible integer for the format. For fixed-length signed integer formats (3), such random integers can be generated using the following pseudocode. In the pseudocode below, INT_MAX is the highest possible integer in the integer format.

METHOD RNDINTRANGE(minInclusive, maxInclusive)
   // minInclusive must not be greater than maxInclusive
   if minInclusive > maxInclusive: return error
   if minInclusive == maxInclusive: return minInclusive
   if minInclusive==0: return RNDINT(maxInclusive)
   // Difference does not exceed maxInclusive
   if minInclusive > 0 or minInclusive + INT_MAX >= maxInclusive
       return minInclusive + RNDINT(maxInclusive - minInclusive)
   end
   while true
     ret = RNDINT(INT_MAX)
     // NOTE: If the signed integer format uses two's-complement
     // form, use the following line:
     if RNDINT(1) == 0: ret = -1 - ret
     // NOTE: If the signed integer format has positive and negative
     // zero, as is the case for Java `float` and
     // `double` and .NET's implementation of `System.Decimal`,
     // for example, use the following
     // three lines instead of the preceding line;
     // here, zero will be rejected at a 50% chance because zero occurs
     // twice in both forms.
     // negative = RNDINT(1) == 0
     // if negative: ret = 0 - ret
     // if negative and ret == 0: continue
     if ret >= minInclusive and ret <= maxInclusive: return ret
   end
END METHOD

Examples:

  • To simulate rolling an N-sided die (N greater than 1), generate a random number in the interval [1, N] by RNDINTRANGE(1, N).
  • Generating a random integer with one base-10 digit is equivalent to generating RNDINTRANGE(0, 9).
  • Generating a random integer with N base-10 digits (where N is 2 or greater) is equivalent to generating RNDINTRANGE(pow(10, N-1), pow(10, N) - 1).

RNDU01(), RNDU01OneExc, RNDU01ZeroExc, and RNDU01ZeroOneExc: Random Numbers Bounded by 0 and 1

This section defines four methods that generate a random number bounded by 0 and 1. In the pseudocode and idioms below, X is the number of maximum even parts, defined later, and INVX is the constant 1 divided by X. For each method below, the alternatives are ordered from most preferred to least preferred.

  • RNDU01(), interval [0, 1]:
    • For Java float or double, use the alternative implementation given later.
    • RNDINT(X) * INVX.
    • RNDINT(X) / X.
  • RNDU01OneExc(), interval [0, 1):

    • Generate RNDU01() in a loop until a number other than 1.0 is generated this way.
    • RNDINT(X - 1) * INVX.
    • RNDINTEXC(X) * INVX.
    • RNDINT(X - 1) / X.
    • RNDINTEXC(X) / X.

    Note that RNDU01OneExc() corresponds to Math.random() in Java and JavaScript. See also "Generating uniform doubles in the unit interval" in the xoroshiro+ remarks page.

  • RNDU01ZeroExc(), interval (0, 1]:
    • Generate RNDU01() in a loop until a number other than 0.0 is generated this way.
    • (RNDINT(X - 1) + 1) * INVX.
    • (RNDINTEXC(X) + 1) * INVX.
    • (RNDINT(X - 1) + 1) / X.
    • (RNDINTEXC(X) + 1) / X.
    • 1.0 - RNDU01OneExc() (but this is recommended only if the set of numbers RNDU01OneExc() could return — as opposed to their probability — is evenly distributed).
  • RNDU01ZeroOneExc(), interval (0, 1):
    • Generate RNDU01() in a loop until a number other than 0.0 or 1.0 is generated this way.
    • (RNDINT(X - 2) + 1) * INVX.
    • (RNDINTEXC(X - 1) + 1) * INVX.
    • (RNDINT(X - 2) + 1) / X.
    • (RNDINTEXC(X - 1) + 1) / X.

Maximum Even Parts

X, the number of maximum even parts, depends on the number format. X is the highest integer p such that p itself and all factors of 1/p between 0 and 1 are representable in that number format. For example—

  • the 64-bit IEEE 754 binary floating-point format (e.g., Java double) has 253 (9007199254740992) maximum even parts,
  • the 32-bit IEEE 754 binary floating-point format (e.g., Java float) has 224 (16777216) maximum even parts,
  • the 64-bit IEEE 754 decimal floating-point format has 1016 maximum even parts, and
  • the .NET Framework decimal format (System.Decimal) would have 296 maximum even parts, but 296 is not representable, so it actually has 295 maximum even parts instead.

Alternative Implementation for RNDU01()

For fixed-precision binary floating-point numbers with fixed exponent range (such as Java's double and float), the following pseudocode for RNDU01() can be used instead. It's based on a technique devised by Allen Downey, who found that dividing a random number by a constant usually does not yield all representable binary floating-point numbers in the desired range. In the pseudocode below, SIGBITS is the binary floating-point format's precision (the number of digits the format can represent without loss; e.g., 53 for Java's double).

METHOD RNDU01()
    e=-SIGBITS
    while true
        if RNDINT(1)==0: e = e - 1
      else: break
    end
    sig = RNDINT((1 << (SIGBITS - 1)) - 1)
    if sig==0 and RNDINT(1)==0: e = e + 1
    sig = sig + (1 << (SIGBITS - 1))
    // NOTE: This multiplication should result in
    // a floating-point number; if `e` is sufficiently
    // small, the number will underflow to 0
    return sig * pow(2, e)
END METHOD

RNDNUMRANGE: Random Numbers in [X, Y]

The naïve way of generating a random number in the interval [minInclusive, maxInclusive], is shown in the following pseudocode, which generally works well only if the number format can't be negative or that format uses arbitrary precision.

METHOD RNDNUMRANGE(minInclusive, maxInclusive)
    if minInclusive > maxInclusive: return error
    return minInclusive + (maxInclusive - minInclusive) * RNDU01()
END

For other number formats (including Java's double and float), the pseudocode above can overflow if the difference between maxInclusive and minInclusive exceeds the maximum possible value for the format. For such formats, the following pseudocode for RNDNUMRANGE() can be used instead. In the pseudocode below, NUM_MAX is the highest possible value for the number format. The pseudocode assumes that the highest possible value is positive and the lowest possible value is negative.

METHOD RNDNUMRANGE(minInclusive, maxInclusive)
   if minInclusive > maxInclusive: return error
   if minInclusive == maxInclusive: return minInclusive
   // Difference does not exceed maxInclusive
   if minInclusive >= 0 or minInclusive + NUM_MAX >= maxInclusive
       return minInclusive + (maxInclusive - minInclusive) * RNDU01()
   end
   while true
     ret = RNDU01() * NUM_MAX
     // NOTE: If the number format has positive and negative
     // zero, as is the case for Java `float` and
     // `double` and .NET's implementation of `System.Decimal`,
     // for example, use the following:
     negative = RNDINT(1) == 0
     if negative: ret = 0 - ret
     if negative and ret == 0: continue
     // NOTE: For fixed-precision fixed-point numbers implemented
     // using two's complement numbers (note 1), use the following line
     // instead of the preceding three lines, where `QUANTUM` is the
     // smallest representable positive number in the fixed-point format:
     // if RNDINT(1) == 0: ret = (0 - QUANTUM) - ret
     if ret >= minInclusive and ret <= maxInclusive: return ret
   end
END

Note: Monte Carlo integration uses randomization to estimate a multidimensional integral. It involves evaluating a function at N random points in the domain. The estimated integral is the volume of the domain times mean of those N points, and the error in the estimate is that volume times the square root of the (bias-corrected sample) variance of the N points (see the appendix). Often quasirandom sequences (also known as low-discrepancy sequences, such as Sobol and Halton sequences), often together with an RNG, provide the "random" numbers to sample the function more efficiently. Unfortunately, the methods to produce such sequences are too complicated to show here.

RNDINTEXC: Random Integers in [0, N)

RNDINTEXC(maxExclusive), which generates a random number in the interval [0, maxExclusive), can be implemented as follows(4):

METHOD RNDINTEXC(maxExclusive)
   if maxExclusive <= 0: return error
   return RNDINT(maxExclusive - 1)
END METHOD

Note: The following are alternative ways of generating a random integer in the interval [**0, maxExclusive):

  • floor(RNDNUMEXCRANGE(0, maxExclusive)).
  • Generate N = floor(RNDU01OneExc()*(maxExclusive)) until N < maxExclusive. (The loop is needed because otherwise, rounding error due to the nature of certain floating-point formats can result in maxExclusive being returned in rare cases.(5))

These approaches, though, are recommended only if the programming language—

  • supports floating-point number types and no other number types (an example is JavaScript),
  • is a dialect of SQL, or
  • doesn't support an integer type that is big enough to fit the number maxExclusive - 1.

RNDINTEXCRANGE: Random Integers in [N, M)

RNDINTEXCRANGE returns a random integer in the interval [minInclusive, maxExclusive). It can be implemented using RNDINTRANGE, as the following pseudocode demonstrates.

METHOD RNDINTEXCRANGE(minInclusive, maxExclusive)
   if minInclusive >= maxExclusive: return error
   // NOTE: For signed integer formats, replace the following line
   // with "if minInclusive >=0 or minInclusive + INT_MAX >=
   // maxExclusive", where `INT_MAX` has the same meaning
   // as the pseudocode for `RNDINTRANGE`.
   if minInclusive >=0
     return RNDINTRANGE(minInclusive, maxExclusive - 1)
   end
   while true
     ret = RNDINTRANGE(minInclusive, maxExclusive)
     if ret < maxExclusive: return ret
   end
END METHOD

RNDNUMEXCRANGE: Random Numbers in [X, Y)

RNDNUMEXCRANGE returns a random number in the interval [minInclusive, maxExclusive). It can be implemented using RNDNUMRANGE, as the following pseudocode demonstrates.

METHOD RNDNUMEXCRANGE(minInclusive, maxExclusive)
   if minInclusive >= maxExclusive: return error
   while true
     ret = RNDNUMRANGE(minInclusive, maxExclusive)
     if ret < maxExclusive: return ret
   end
END METHOD

Uniform Random Bits

The idiom RNDINT((1 << b) - 1) is a naïve way of generating a uniform random N-bit integer (with maximum 2b - 1).

In practice, memory is usually divided into bytes, or 8-bit unsigned integers in the interval [0, 255]. In this case, a byte array or a block of memory can be filled with random bits by setting each byte to RNDINT(255). (There may be faster, RNG-specific ways to fill memory with random bytes, such as with RNGs that generate random numbers in parallel. These ways are not detailed in this document.)

Special Programming Environments

In certain programming environments it's often impractical to implement the uniform random number generation methods just described without recurring to other programming languages. For instance:

  • Shell scripts and Microsoft Windows batch files are designed for running other programs, rather than general-purpose programming. However, batch files and bash (a shell script interpreter) might support a variable which returns a random integer in the interval [0, 65535] (called %RANDOM% and $RANDOM, respectively); neither variable is designed for information security.
  • Standard SQL does not include an RNG in its suite of functionality, but popular SQL dialects often do — with idiosyncratic behavior.(6)
  • In functional programming languages such as Haskell, it may be important to separate code that directly uses RNGs from other code, usually by rewriting certain functions to take one or more pregenerated random numbers rather than directly using RNDINT, RNDNUMRANGE, or another random number generation method presented earlier in this document.

Randomization Techniques

This section describes commonly used randomization techniques, such as shuffling, selection of several unique items, and creating random strings of text.

Boolean Conditions

To generate a condition that is true at the specified probabilities, use the following idioms in an if condition:

  • True or false with equal probability: RNDINT(1) == 0.
  • True with X percent probability: RNDINTEXC(100) < X.
  • True with probability X/Y: RNDINTEXC(Y) < X.
  • True with odds of X to Y: RNDINTEXC(X + Y) < X.
  • True with probability X, where X is from 0 through 1 (a Bernoulli trial): RNDU01OneExc() < X.

Examples:

  • True with probability 3/8: RNDINTEXC(8) < 3.
  • True with odds of 100 to 1: RNDINTEXC(101) < 1.
  • True with 20% probability: RNDINTEXC(100) < 20.

Shuffling

The Fisher–Yates shuffle method shuffles a list (puts its items in a random order) such that all permutations (arrangements) of that list are equally likely to occur, assuming the RNG it uses can choose any one of those permutations. However, that method is also easy to write incorrectly (see also Jeff Atwood, "The danger of naïveté"). The following pseudocode is designed to shuffle a list's contents.

METHOD Shuffle(list)
   // NOTE: Check size of the list early to prevent
   // `i` from being less than 0 if the list's size is 0 and
   // `i` is implemented using an unsigned type available
   // in certain programming languages.
   if size(list) >= 2
      // Set i to the last item's index
      i = size(list) - 1
      while i > 0
         // Choose an item ranging from the first item
         // up to the item given in `i`. Note that the item
         // at i+1 is excluded.
         k = RNDINTEXC(i + 1)
         // The following is wrong since it introduces biases:
         // "k = RNDINTEXC(size(list))"
         // The following is wrong since the algorithm won't
         // choose from among all possible permutations:
         // "k = RNDINTEXC(i)"
         // Swap item at index i with item at index k;
         // in this case, i and k may be the same
         tmp = list[i]
         list[i] = list[k]
         list[k] = tmp
         // Move i so it points to the previous item
         i = i - 1
      end
   end
   // NOTE: An implementation can return the
   // shuffled list, as is done here, but this is not required.
   return list
END METHOD

An important consideration with respect to shuffling is the nature of the underlying RNG, as I discuss in further detail in my RNG recommendation document on shuffling.(7)

Note: In simulation testing, shuffling is used to relabel items from a dataset at random, where each item in the dataset is assigned one of several labels. In such testing—

  • one or more statistics that involve the specific labeling of the original dataset's groups is calculated (such as the difference, maximum, or minimum of means or variances between groups), then
  • multiple simulated datasets are generated, where each dataset is generated by—
    • merging the groups,
    • shuffling the merged dataset, and
    • relabeling each item in order such that the number of items in each group for the simulated dataset is the same as for the original dataset, then
  • for each simulated dataset, the same statistics are calculated as for the original dataset, then
  • the statistics for the simulated datasets are compared with those of the original.

Sampling With Replacement: Choosing a Random Item from a List

To choose a random item from a list—

  • whose size is known in advance, use the idiom list[RNDINTEXC(size(list))]; or
  • whose size is not known in advance, generate RandomKItemsFromFile(file, 1), in pseudocode given in a later section (the result will be a 1-item list or be an empty list if there are no items).

Choosing an item this way is also known as sampling with replacement.

Notes:

  • Generating a random number in the interval [mn, mx) in increments equal to step is equivalent to—
    • generating a list of all numbers in the interval [mn, mx) of the form mn + step * x, where x >= 0 is an integer, then
    • choosing a random item from the list generated this way.
  • Bootstrapping is a method of creating a simulated dataset by choosing random items with replacement from an existing dataset until both datasets have the same size. (The simulated dataset can contain duplicates this way.) Usually, multiple simulated datasets are generated this way, one or more statistics, such as the mean, are calculated for each simulated dataset as well as the original dataset, and the statistics for the simulated datasets are compared with those of the original.

Example: Random Character Strings

To generate a random string of characters:

  1. Generate a list of the letters, digits, and/or other characters the string can have. Examples are given later in this section.
  2. Build a new string whose characters are chosen from that character list. The pseudocode below demonstrates this by creating a list, rather than a string, where the random characters will be held. It also takes the number of characters as a parameter named size. (How to convert this list to a text string depends on the programming language and is outside the scope of this page.)

    METHOD RandomString(characterList, stringSize)
            i = 0
            newString = NewList()
            while i < stringSize
                    // Choose a character from the list
                    randomChar = characterList[RNDINTEXC(size(characterList))]
                    // Add the character to the string
                    AddItem(newString, randomChar)
                    i = i + 1
            end
            return newString
    END METHOD

Notes:

  1. If the list of characters is fixed, the list can be statically created at runtime or compile time, or a string type as provided in the programming language can be used to store the list as a string.
  2. Instead of individual characters, the list can consist of strings of one or more characters each (e.g., words or syllables), or indeed any other items. (In that case, the individual strings or items should not be stored as a single string).
  3. Often applications need to generate a string of characters that's not only random, but also unique. This can be done by storing a list (such as a hash table) of strings already generated and checking newly generated strings against that list. If the strings identify database records, file system paths, or other shared resources, special considerations apply, including the need to synchronize access, but are not discussed further in this document.
  4. Generating "pronounceable" words or words similar to natural-language words is generally more sophisticated than shown above. Often, doing so involves Markov chains. A Markov chain models one or more states (for example, individual letters or syllables), and stores the probabilities to transition between these states (e.g., "b" to "e" with a probability of 0.2, or "b" to "b" with a probability of 0.01). A Markov chain modeling random word generation, for example, can include "start" and "stop" states for the start and end of the word, respectively.

Examples of character lists:

  1. For an alphanumeric string, or string of letters and digits, the characters can be the basic digits "0" to "9" (U+0030-U+0039, nos. 48-57), the basic upper case letters "A" to "Z" (U+0041-U+005A, nos. 65-90), and the basic lower case letters "a" to "z" (U+0061-U+007A, nos. 96-122), as given in the Unicode Standard.
  2. For a base-10 digit string, the characters can be the basic digits only.
  3. For a base-16 digit (hexadecimal) string, the characters can be the basic digits as well as the basic letters "A" to "F" or "a" to "f".

Sampling Without Replacement: Choosing Several Unique Items

There are several techniques for choosing k unique items or values uniformly at random from among n available items or values, depending on such things as whether n is known and how big n and k are.

  1. If n is not known in advance: Use the reservoir sampling method; see the RandomKItemsFromFile method in the pseudocode below. Although the pseudocode refers to files and lines, the technique applies to any situation when items are retrieved one at a time from a dataset or list whose size is not known in advance.
  2. If items are to be chosen in order:
    • If n is relatively small, then the RandomKItemsInOrder method, in the pseudocode below, demonstrates a solution (based on a technique presented in Devroye 1986, p. 620).
    • If n is relatively large, see the item "If n is relatively large", later.
  3. If n is relatively small (for example, if there are 200 available items, or there is a range of numbers from 0 to 200 to choose from): Do one of the following:
    • Store all the items in a list, shuffle that list, then choose the first k items from that list.
    • If the items are already stored in a list and the list's order can be changed, then shuffle that list and choose the first k items from the shuffled list.
    • If the items are already stored in a list and the list's order can't be changed, then store the indices to those items in another list, shuffle the latter list, then choose the first k indices (or the items corresponding to those indices) from the latter list.
  4. If k is much smaller than n and the items are stored in a list whose order can be changed: Do a partial shuffle of that list, then choose the last k items from that list. A partial shuffle proceeds as given in the section "Shuffling", except the partial shuffle stops after k swaps have been made (where swapping one item with itself counts as a swap).
  5. If k is much smaller than n and n is not very large (for example, less than 5000): Do one of the following:
    • Store all the items in a list, do a partial shuffle of that list, then choose the last k items from that list.
    • If the items are already stored in a list and the list's order can't be changed, then store the indices to those items in another list, do a partial shuffle of the latter list, then choose the last k indices (or the items corresponding to those indices) from the latter list.
  6. If n - k is much smaller than n, and the order in which the items are sampled need not be random:
    • If the items are stored in a list whose order can be changed, then proceed as in step 4, except the partial shuffle involves n - k swaps and the first k items are chosen rather than the last k.
    • Otherwise, if n is not very large, then proceed as in step 5, except the partial shuffle involves n - k swaps and the first k items or indices are chosen rather than the last k.
  7. Otherwise (for example, if 32-bit or larger integers will be chosen so that n is 232 or is n is otherwise very large): Create a hash table storing the indices to items already chosen. When a new index to an item is randomly chosen, check the hash table to see if it's there already. If it's not there already, add it to the hash table. Otherwise, choose a new random index. Repeat this process until k indices were added to the hash table this way. If the items are to be chosen in order, then a red–black tree, rather than a hash table, can be used to store the indices this way; after k indices are added to the tree, the indices (and the items corresponding to them) can be retrieved in sorted order. J. Preshing discusses different ways to generate unique random integers.

Choosing several unique items as just described is also known as sampling without replacement.

The following pseudocode implements the RandomKItemsFromFile and RandomKItemsInOrder methods referred to in this section.

 METHOD RandomKItemsFromFile(file, k)
    list = NewList()
    j = 0
    endOfFile = false
    while j < k
       // Get the next line from the file
       item = GetNextLine(file)
       // The end of the file was reached, break
       if item == nothing:
          endOfFile = true
          break
       end
       AddItem(list, item)
       j = j + 1
    end
    i = 1 + k
    while endOfFile == false
       // Get the next line from the file
       item = GetNextLine(file)
       // The end of the file was reached, break
       if item == nothing: break
       j = RNDINTEXC(i)
       if j < k: list[j] = item
       i = i + 1
    end
    // We shuffle at the end in case k or fewer
    // lines were in the file, since in that
    // case the items would appear in the same
    // order as they appeared in the file
    // if the list weren't shuffled.  This line
    // can be removed, however, if the items
    // in the returned list need not appear
    // in random order.
    Shuffle(list)
    return list
 end

METHOD RandomKItemsInOrder(list, k)
        i = 0
        kk = k
        ret = NewList()
        n = size(list)
        while i  < n and size(ret) < k
          u = RNDINTEXC(n - i)
          if u <= kk
           AddItem(ret, list[i])
           kk = kk - 1
          end
          i = i + 1
       end
       return ret
 END METHOD

Note: Removing k random items from a list of n items (list) is equivalent to generating a new list by RandomKItemsInOrder(list, n - k).

Choosing a Random Date/Time

Choosing a random date/time at or between two others is equivalent to—

  • converting the two input date/times to an integer or number (here called date1 and date2, where date1 represents the earlier date/time and date2 the other) at the required granularity, for instance, month, day, or hour granularity (the details of such conversion depend on the date/time format and are outside the scope of this document),
  • generating newDate = RNDINTRANGE(date1, date2) or newDate = RNDNUMRANGE(date1, date2), respectively, and
  • converting newDate to a date/time.

If either input date/time was generated as the random date, but that is not desired, the process just given can be repeated until such a date/time is not generated this way.

Generating Random Numbers in Sorted Order

The following pseudocode describes a method that generates random numbers in the interval [0, 1] in sorted order. count is the number of random numbers to generate this way. The method is based on an algorithm from Bentley and Saxe 1979.

METHOD SortedRandom(count)
   list = NewList()
   k = count
   c = 1.0
   while k > 0
       c = pow(RNDU01(), 1.0 / k) * c
       AddItem(list, c)
   end
   return list
END METHOD

Alternatively, random numbers can be generated (using any method and where the numbers have any distribution and range) and stored in a list, and the list then sorted using a sorting algorithm. Details on sorting algorithms, however, are beyond the scope of this document.

Rejection Sampling

Rejection sampling is a simple and flexible approach for generating random content that meets certain requirements. To implement rejection sampling:

  1. Generate the random content (such as a random number) by any method and with any distribution and range.
  2. If the content doesn't meet predetermined criteria, go to step 1.

Example criteria include checking any one or more of—

  • whether a random number is prime,
  • whether a random number is divisible or not by certain numbers,
  • whether a random number is not among recently chosen random numbers,
  • whether a random number was not already chosen (with the aid of a hash table, red-black tree, or similar structure),
  • whether a random point is sufficiently distant from previous random points (with the aid of a KD-tree or similar structure),
  • whether a random string matches a regular expression, and
  • whether a random number is not included in a "blacklist" of numbers.

(KD-trees, hash tables, red-black trees, prime-number testing algorithms, and regular expressions are outside the scope of this document.)

Random Walks

A random walk is a process with random behavior over time. A simple form of random walk involves generating a random number that changes the state of the walk. The pseudocode below generates a random walk of n random numbers, where STATEJUMP() is the next number to add to the current state (see examples later in this section).

METHOD RandomWalk(n)
  // Create a new list with an initial state
  list=[0]
  // Add 'n' new numbers to the list.
  for i in 0...n: AddItem(list, list[i] + STATEJUMP())
  return list
END METHOD

Examples:

  1. If STATEJUMP() is RNDINT(1) * 2 - 1, the random walk generates numbers that differ by -1 or 1, chosen at random.
  2. If STATEJUMP() is RNDNUMRANGE(-1, 1), the random state is advanced by a random real number in the interval [-1, 1].
  3. If STATEJUMP() is Binomial(1, p), the random walk models a binomial process, where the state is advanced with probability p.
  4. If STATEJUMP() is Binomial(1, p) * 2 - 1, the random walk generates numbers that each differ from the last by -1 or 1 depending on the probability p.

Notes:

  1. A random process can also be simulated by creating a list of random numbers generated the same way. Such a process generally models behavior over time that does not depend on the time or the current state. Examples of this include Normal(0, 1) (for modeling Gaussian white noise) and Binomial(1, p) (for modeling a Bernoulli process, where each number is 0 or 1 depending on the probability p).
  2. Some random walks model random behavior at every moment, not just at discrete times. One example is a Wiener process, with random states and jumps that are normally distributed (a process of this kind is also known as Brownian motion). (For a random walk that follows a Wiener process, STATEJUMP() is Normal(mu * timediff, sigma * sqrt(timediff)), where mu is the average value per time unit, sigma is the volatility, and timediff is the time difference between samples.)
  3. Some random walks model state changes happening at random times. One example is a Poisson process, in which the time between each event is a random exponential variable (the random time is -ln(RNDU01ZeroOneExc()) / rate, where rate is the average number of events per time unit; an inhomogeneous Poisson process results if rate can vary with the "timestamp" before each event jump).

General Non-Uniform Distributions

Some applications need to choose random items or numbers such that some of them are more likely to be chosen than others (a non-uniform distribution). Most of the techniques in this section show how to use the uniform random number methods to generate such random items or numbers.

Weighted Choice

The weighted choice method generates a random item from among a collection of them with separate probabilities of each item being chosen.

The following pseudocode takes a single list weights, and returns the index of a weight from that list. The greater the weight, the more likely its index will be chosen. (Note that there are two possible ways to generate the random number depending on whether the weights are all integers or can be fractional numbers.) Each weight should be 0 or greater.

METHOD WeightedChoice(weights)
    if size(weights) == 0: return error
    sum = 0
    // Get the sum of all weights
    i = 0
    while i < size(weights)
        sum = sum + weights[i]
        i = i + 1
    end
    // Choose a random integer/number from 0 to less than
    // the sum of weights.
    value = RNDINTEXC(sum)
    // NOTE: If the weights can be fractional numbers,
    // use this instead:
    // value = RNDNUMEXCRANGE(0, sum)
    // Choose the object according to the given value
    i = 0
    lastItem = size(weights) - 1
    runningValue = 0
    while i < size(weights)
       if weights[i] > 0
          newValue = runningValue + weights[i]
          // NOTE: Includes start, excludes end
          if value < newValue: return i
          runningValue = newValue
          lastItem = i
       end
       i = i + 1
    end
    // Last resort (might happen because rounding
    // error happened somehow)
    return lastItem
END METHOD

Examples:

  1. Assume we have the following list: ["apples", "oranges", "bananas", "grapes"], and weights is the following: [3, 15, 1, 2]. The weight for "apples" is 3, and the weight for "oranges" is 15. Since "oranges" has a higher weight than "apples", the index for "oranges" (1) is more likely to be chosen than the index for "apples" (0) with the WeightedChoice method. The following pseudocode implements how to get a randomly chosen item from the list with that method.

    index = WeightedChoice(weights)
    // Get the actual item
    item = list[index]
    
  2. Assume the weights from example 1 are used and the list contains ranges of numbers instead of strings: [[0, 5], [5, 10], [10, 11], [11, 13]]. If a random range is chosen, a random number can be chosen from that range using code like the following: number = RNDNUMEXCRANGE(item[0], item[1]). (See also "Mixtures of Distributions".)

  3. Assume the weights from example 1 are used and the list contains the following: [0, 5, 10, 11, 13] (one more item than the weights). This expresses four ranges, the same as in example 2. After a random index is chosen with index = WeightedChoice(weights), a random number can be chosen from the corresponding range using code like the following: number = RNDNUMEXCRANGE(list[index], list[index + 1]). (This is how the C++ library expresses a piecewise constant distribution.)

In all the examples above, the weights sum to 21. However, when 21 items are selected, the index for "apples" will not necessarily be chosen exactly 3 times, or the index for "oranges" exactly 15 times, for example. Each number generated by WeightedChoice is independent from the others, and each weight indicates only a likelihood that the corresponding index will be chosen rather than the other indices. And this likelihood doesn't change no matter how many times WeightedChoice is given the same weights. This is called a weighted choice with replacement, which can be thought of as drawing a ball, then putting it back.

Weighted Choice Without Replacement (Multiple Copies)

To implement weighted choice without replacement (which can be thought of as drawing a ball without putting it back), generate an index by WeightedChoice, and then decrease the weight for the chosen index by 1. In this way, each weight behaves like the number of "copies" of each item. This technique, though, will only work properly if all the weights are integers 0 or greater. The pseudocode below is an example of this.

// Get the sum of weights
// (NOTE: This code assumes that `weights` is
// a list that can be modified.  If the original weights
// are needed for something else, a copy of that
// list should be made first, but the copying process
// is not shown here.  This code also assumes that `list`,
// a list of items, was already declared earlier and
// has at least as many items as `weights`.)
totalWeight = 0
i = 0
while i < size(weights)
    totalWeight = totalWeight + weights[i]
    i = i + 1
end
// Choose as many items as the sum of weights
i = 0
items = NewList()
while i < totalWeight
    index = WeightedChoice(weights)
    // Decrease weight by 1 to implement selection
    // without replacement.
    weights[index] = weights[index] - 1
    AddItem(items, list[index])
    i = i + 1
end

Alternatively, if all the weights are integers 0 or greater and their sum is relatively small, create a list with as many copies of each item as its weight, then shuffle that list. The resulting list will be ordered in a way that corresponds to a weighted random choice without replacement.

Note: Weighted choice without replacement can be useful to some applications (particularly some games) that wish to control which random numbers appear, to make the random outcomes appear fairer to users (e.g., to avoid long streaks of good outcomes or of bad outcomes). When used for this purpose, each item represents a different outcome (e.g., "good" or "bad"), and the lists are replenished once no further items can be chosen. However, this kind of sampling should not be used for this purpose whenever information security (ISO/IEC 27000) is involved, including when predicting future random numbers would give a player or user a significant and unfair advantage.

Weighted Choice Without Replacement (Single Copies)

Weighted choice can also choose items from a list, where each item has a separate probability of being chosen and can be chosen no more than once. In this case, after choosing a random index, set the weight for that index to 0 to keep it from being chosen again. The pseudocode below is an example of this.

// (NOTE: This code assumes that `weights` is
// a list that can be modified.  If the original weights
// are needed for something else, a copy of that
// list should be made first, but the copying process
// is not shown here.  This code also assumes that `list`,
// a list of items, was already declared earlier and
// has at least as many items as `weights`.)
chosenItems = NewList()
i = 0
// Choose k items from the list
while i < k or i < size(weights)
    index = WeightedChoice(weights)
    // Set the weight for the chosen index to 0
    // so it won't be chosen again
    weights[index] = 0
    // Add the item at the chosen index
    AddItem(chosenItems, list[index])
end
// `chosenItems` now contains the items chosen

The technique presented here can solve the problem of sorting a list of items such that higher-weighted items are more likely to appear first.

Continuous Weighted Choice

The continuous weighted choice method generates a random number that follows a continuous probability distribution (here, a piecewise linear distribution).

The pseudocode below takes two lists as follows:

  • list is a list of numbers (which need not be integers). If the numbers are arranged in ascending order, which they should, the first number in this list can be returned exactly, but not the last number.
  • weights is a list of weights for the given numbers (where each number and its weight have the same index in both lists). The greater a number's weight, the more likely it is that a number close to that number will be chosen. Each weight should be 0 or greater.

 

METHOD ContinuousWeightedChoice(list, weights)
    if size(list) <= 0 or size(weights) < size(list): return error
    if size(list) == 1: return list[0]
    // Get the sum of all areas between weights
    sum = 0
    areas = NewList()
    i = 0
    while i < size(list) - 1
      weightArea = abs((weights[i] + weights[i + 1]) * 0.5 *
            (list[i + 1] - list[i]))
      AddItem(areas, weightArea)
      sum = sum + weightArea
       i = i + 1
    end
    // Choose a random number
    value = RNDNUMEXCRANGE(0, sum)
    // Interpolate a number according to the given value
    i=0
    // Get the number corresponding to the random number
    runningValue = 0
    while i < size(list) - 1
     area = areas[i]
     if area > 0
      newValue = runningValue + area
      // NOTE: Includes start, excludes end
      if value < newValue
       // NOTE: The following line can also read
       // "interp = RNDU01OneExc()", that is, a new number is generated
       // within the chosen area rather than using the point
       // already generated.
       interp = (value - runningValue) / (newValue - runningValue)
       retValue = list[i] + (list[i + 1] - list[i]) * interp
       return retValue
      end
      runningValue = newValue
     end
     i = i + 1
    end
    // Last resort (might happen because rounding
    // error happened somehow)
    return list[size(list) - 1]
END METHOD

Example: Assume list is the following: [0, 1, 2, 2.5, 3], and weights is the following: [0.2, 0.8, 0.5, 0.3, 0.1]. The weight for 2 is 0.5, and that for 2.5 is 0.3. Since 2 has a higher weight than 2.5, numbers near 2 are more likely to be chosen than numbers near 2.5 with the ContinuousWeightedChoice method.

Mixtures of Distributions

A mixture consists of two or more probability distributions with separate probabilities of being sampled. To generate random content from a mixture—

  1. generate index = WeightedChoice(weights), where weights is a list of relative probabilities that each distribution in the mixture will be sampled, then
  2. based on the value of index, generate the random content from the corresponding distribution.

Examples:

  1. One mixture consists of two normal distributions with two different means: 1 and -1, but the mean 1 normal will be sampled 80% of the time. The following pseudocode shows how this mixture can be sampled:

    index = WeightedChoice([80, 20])
    number = 0
    // If index 0 was chosen, sample from the mean 1 normal
    if index==0: number = Normal(1, 1)
    // Else index 1 was chosen, so sample from the mean -1 normal
    else: number = Normal(-1, 1)
  2. A hyperexponential distribution is a mixture of exponential distributions, each one with a separate weight and separate rate. An example is below.

    index = WeightedChoice([0.6, 0.3, 0.1])
    // Rates of the three exponential distributions
    rates = [0.3, 0.1, 0.05]
    // Generate an exponential random number with chosen rate
    number = -ln(RNDU01ZeroOneExc()) / rates[index]
  3. Choosing a point uniformly at random from a complex shape (in any number of dimensions) is equivalent to sampling uniformly from a mixture of simpler shapes that make up the complex shape (here, the weights list holds the content of each simpler shape). (Content is called area in 2D and volume in 3D.) For example, a simple closed 2D polygon can be triangulated, or decomposed into triangles, and a mixture of those triangles can be sampled.(8)

  4. For generating a random integer from multiple nonoverlapping ranges of integers—

    • each range has a weight of (mx - mn) + 1, where mn is that range's minimum and mx is its maximum, and
    • the chosen range is sampled by generating RNDINTRANGE(mn, mx), where mn is the that range's minimum and mx is its maximum.

    For generating random numbers, that may or may not be integers, from nonoverlapping number ranges, each weight is mx - mn instead and the number is sampled by RNDNUMEXCRANGE(mn, mx) instead.

Random Numbers from a Distribution of Data Points

Generating random numbers (or data points) based on how a list of numbers (or data points) is distributed involves a family of techniques called density estimation, which include histograms, kernel density estimation, and Gaussian mixture models. These techniques seek to model the distribution of data points in a given data set, where areas with more points are more likely to be sampled.

  1. Histograms are sets of one or more bins, which are generally of equal size. Histograms are mixtures, where each bin's weight is the number of data points in that bin. After a bin is randomly chosen, a random data point that could fit in that bin is generated (that point need not be an existing data point).
  2. Gaussian mixture models are also mixtures, in this case, mixtures of one or more Gaussian (normal) distributions.
  3. Kernel distributions are mixtures of sampling distributions, one for each data point. Estimating a kernel distribution is called kernel density estimation. To sample from a kernel distribution:
    1. Choose one of the numbers or points in the list at random with replacement.
    2. Add a randomized "jitter" to the chosen number or point; for example, add a separately generated Normal(0, sigma) to the chosen number or each component of the chosen point, where sigma is the bandwidth(9).

This document doesn't detail how to build a density estimation model. Other references on density estimation include a Wikipedia article on multiple-variable kernel density estimation, and a blog post by M. Kay.

Transformations of Random Numbers

Random numbers can be generated by combining and/or transforming one or more random numbers and/or discarding some of them.

As an example, "Probability and Games: Damage Rolls" by Red Blob Games includes interactive graphics showing score distributions for lowest-of, highest-of, drop-the-lowest, and reroll game mechanics.(10) These and similar distributions can be generalized as follows.

Generate two or more random numbers, each with a separate probability distribution, then:

  1. Highest-of: Choose the highest generated number.
  2. Drop-the-lowest: Add all generated numbers except the lowest.
  3. Reroll-the-lowest: Add all generated numbers except the lowest, then add a number generated randomly by a separate probability distribution.
  4. Lowest-of: Choose the lowest generated number.
  5. Drop-the-highest: Add all generated numbers except the highest.
  6. Reroll-the-highest: Add all generated numbers except the highest, then add a number generated randomly by a separate probability distribution.
  7. Sum: Add all generated numbers.
  8. Mean: Find the mean of all generated numbers (see the appendix).

If the probability distributions are the same, then strategies 1 to 3 make higher numbers more likely, and strategies 4 to 6, lower numbers.

Notes: Variants of strategy 4 — e.g., choosing the second-, third-, or nth-lowest number — are formally called second-, third-, or nth-order statistics distributions, respectively.

Examples:

  1. The idiom min(RNDINTRANGE(1, 6), RNDINTRANGE(1, 6)) takes the lowest of two six-sided die results. Due to this approach, 1 is more likely to occur than 6.
  2. The idiom RNDINTRANGE(1, 6) + RNDINTRANGE(1, 6) takes the result of two six-sided dice (see also "Dice").
  3. Sampling a Bates distribution involves sampling n random numbers by RNDNUMRANGE(minimum, maximum), then finding the mean of those numbers (see the appendix).
  4. A compound Poisson distribution models the sum of n random numbers each generated the same way, where n follows a Poisson distribution (e.g., n = Poisson(10) for an average of 10 numbers).
  5. A hypoexponential distribution models the sum of n random numbers following an exponential distribution, each with a separate lamda parameter (see "Gamma Distribution").

Random Numbers from an Arbitrary Distribution

Many probability distributions can be defined in terms of any of the following:

  • The cumulative distribution function, or CDF, returns, for each number, the probability for a randomly generated variable to be equal to or less than that number; the probability is in the interval [0, 1].
  • The probability density function, or PDF, is, roughly and intuitively, a curve of weights 0 or greater, where for each number, the greater its weight, the more likely a number close to that number is randomly chosen.(11)

If a probability distribution's PDF is known, one of the following techniques, among others, can be used to generate random numbers that follow that distribution.

  1. Use the PDF to calculate the weights for a number of sample points (usually regularly spaced). Create one list with the sampled points in ascending order (the list) and another list of the same size with the PDF's values at those points (the weights). Finally generate ContinuousWeightedChoice(list, weights) to generate a random number bounded by the lowest and highest sampled point. This technique can be used even if the area under the PDF isn't 1. OR
  2. Use rejection sampling. Choose the lowest and highest random number to generate (minValue and maxValue, respectively) and find the maximum value of the PDF at or between those points (maxDensity). The rejection sampling approach is then illustrated with the following pseudocode, where PDF(X) is the distribution's PDF (see also Saucier 2000, p. 39). This technique can be used even if the area under the PDF isn't 1.

    METHOD ArbitraryDist(minValue, maxValue, maxDensity)
         if minValue >= maxValue: return error
         while True
             x=RNDNUMEXCRANGE(minValue, maxValue)
             y=RNDNUMEXCRANGE(0, maxDensity)
             if y < PDF(x): return x
         end
    END METHOD

If both a PDF and a uniform random variable in the interval [0, 1) (randomVariable) are given, then the following technique, among other possible techniques, can be used: Do the same process as method 1, given earlier, except—

  • divide the weights in the weights list by the sum of all weights, and
  • use a modified version of ContinuousWeightedChoice that uses randomVariable rather than generating a new random number.

If the distribution's CDF is known, an approach called inverse transform sampling can be used: Generate ICDF(RNDU01ZeroOneExc()), where ICDF(X) is the distribution's inverse CDF. The Python sample code includes from_interp and numbers_from_cdf methods that implement this approach numerically.

Note: Further details on inverse transform sampling or on how to find inverses, as well as lists of PDFs and CDFs, are outside the scope of this page.

Censored and Truncated Distributions

To sample from a censored probability distribution, generate a random number from that distribution and—

  • if that number is less than a minimum threshold, use the minimum threshold instead, and/or
  • if that number is greater than a maximum threshold, use the maximum threshold instead.

To sample from a truncated probability distribution, generate a random number from that distribution and, if that number is less than a minimum threshold and/or higher than a maximum threshold, repeat this process.

Correlated Random Numbers

According to (Saucier 2000), sec. 3.8, to generate two correlated (dependent) random variables—

  • generate two independent and identically distributed random variables x and y (for example, two Normal(0, 1) variables or two RNDU01() variables), and
  • calculate [x, y*sqrt(1 - rho * rho) + rho * x], where rho is a correlation coefficient in the interval [-1, 1] (if rho is 0, the variables are uncorrelated).

Other ways to generate correlated random numbers are explained in the section "Gaussian and Other Copulas".

Specific Non-Uniform Distributions

This section contains information on some of the most common non-uniform probability distributions.

Dice

The following method generates a random result of rolling virtual dice.(12) It takes three parameters: the number of dice (dice), the number of sides in each die (sides), and a number to add to the result (bonus) (which can be negative, but the result of the subtraction is 0 if that result is greater).

METHOD DiceRoll(dice, sides, bonus)
    if dice < 0 or sides < 1: return error
    if dice == 0: return 0
    if sides == 1: return dice
    ret = 0
    if dice > 50
        // If there are many dice to roll,
        // use a faster approach, noting that
        // the dice-roll distribution approaches
        // a "discrete" normal distribution as the
        // number of dice increases.
        mean = dice * (sides + 1) * 0.5
        sigma = sqrt(dice * (sides * sides - 1) / 12)
        ret = -1
        while ret < dice or ret > dice * sides
            ret = round(Normal(mean, sigma))
        end
    else
         i = 0
         while i < dice
              ret = ret + RNDINTRANGE(1, sides)
              i = i + 1
          end
    end
    ret = ret + bonus
    if ret < 0: ret = 0
    return ret
END METHOD

Examples: The result of rolling—

  • four six-sided virtual dice ("4d6") is DiceRoll(4,6,0),
  • three ten-sided virtual dice, with 4 added ("3d10 + 4"), is DiceRoll(3,10,4), and
  • two six-sided virtual dice, with 2 subtracted ("2d6 - 2"), is DiceRoll(2,6,-2).

Normal (Gaussian) Distribution

The normal distribution (also called the Gaussian distribution) can be implemented using the pseudocode below, which uses the polar method (13) to generate two normally-distributed random numbers:

  • mu (μ) is the mean (average), or where the peak of the distribution's "bell curve" is.
  • sigma (σ), the standard deviation, affects how wide the "bell curve" appears. The probability that a normally-distributed random number will be within one standard deviation from the mean is about 68.3%; within two standard deviations (2 times sigma), about 95.4%; and within three standard deviations, about 99.7%.

 

METHOD Normal2(mu, sigma)
  while true
    a = RNDU01()
    b = RNDU01()
    if a != 0 and RNDINT(1) == 0: a = 0 - a
    if b != 0 and RNDINT(1) == 0: b = 0 - b
    c = a * a + b * b
    if c != 0 and c <= 1
       c = sqrt(-2 * ln(c) / c)
       return [a * mu * c + sigma, b * mu * c + sigma]
    end
  end
END METHOD

Since Normal2 returns two numbers instead of one, but many applications require only one number at a time, a problem arises on how to return one number while storing the other for later retrieval. Ways to solve this problem are outside the scope of this page, however. The name Normal will be used in this document to represent a method that returns only one normally-distributed random number rather than two.

Alternatively, or in addition, the following method (implementing a ratio-of-uniforms technique) can be used to generate normally distributed random numbers.

METHOD Normal(mu, sigma)
    bmp = sqrt(2.0/exp(1.0)) // about 0.8577638849607068
    while true
        a=RNDU01ZeroExc()
        b=RNDNUMRANGE(-bmp,bmp)
        if b*b <= -a * a * 4 * ln(a)
            return (b * sigma / a) + mu
        end
    end
END METHOD

Notes:

  • In a standard normal distribution, mu = 0 and sigma = 1.
  • If variance is given, rather than standard deviation, the standard deviation (sigma) is the variance's square root.

Binomial Distribution

A random integer that follows a binomial distribution

  • expresses the number of successes that have happened after a given number of independently performed trials (expressed as trials below), where the probability of a success in each trial is p (which ranges from 0, never, to 1, always, and which can be 0.5, meaning an equal chance of success or failure), and
  • is also known as Hamming distance, if each trial is treated as a "bit" that's set to 1 for a success and 0 for a failure, and if p is 0.5.

 

METHOD Binomial(trials, p)
    if trials < 0: return error
    if trials == 0: return 0
    // Always succeeds
    if p >= 1.0: return trials
    // Always fails
    if p <= 0.0: return 0
    i = 0
    count = 0
    // Suggested by Saucier, R. in "Computer
    // generation of probability distributions", 2000, p. 49
    tp = trials * p
    if tp > 25 or (tp > 5 and p > 0.1 and p < 0.9)
         countval = -1
         while countval < 0 or countval > trials
              countval = round(Normal(tp, tp))
         end
         return countval
    end
    if p == 0.5
        while i < trials
            if RNDINT(1) == 0
                // Success
                count = count + 1
            end
            i = i + 1
        end
    else
        while i < trials
            if RNDU01OneExc() < p
                // Success
                count = count + 1
            end
            i = i + 1
        end
    end
    return count
END METHOD

Examples:

  • If p is 0.5, the binomial distribution models the task "Flip N coins, then count the number of heads."
  • The idiom Binomial(N, 0.5) >= C is true if at least C coins, among N coins flipped, show the successful outcome (for example, heads if heads is the successful outcome).
  • The idiom Binomial(N, 1/S) models the task "Roll N S-sided dice, then count the number of dice that show the number S."

Poisson Distribution

The following method generates a random integer that follows a Poisson distribution and is based on Knuth's method from 1969. In the method—

  • mean is the average number of independent events of a certain kind per fixed unit of time or space (for example, per day, hour, or square kilometer), and can be an integer or a non-integer (the method allows mean to be 0 mainly for convenience), and
  • the method's return value gives a random number of such events within one such unit.

 

METHOD Poisson(mean)
    if mean < 0: return error
    if mean == 0: return 0
    p = 1.0
    // Suggested by Saucier, R. in "Computer
    // generation of probability distributions", 2000, p. 49
    if mean > 9
        p = -1.0
        while p < 0: p = round(Normal(mean, mean))
        return p
    end
    pn = exp(-mean)
    count = 0
    while true
        p = p * RNDU01OneExc()
        if p <= pn: return count
        count = count + 1
    end
END METHOD

Gamma Distribution

The following method generates a random number that follows a gamma distribution and is based on Marsaglia and Tsang's method from 2000. Usually, the number expresses either—

  • the lifetime (in days, hours, or other fixed units) of a random component with an average lifetime of meanLifetime, or
  • a random amount of time (in days, hours, or other fixed units) that passes until as many events as meanLifetime happen.

Here, meanLifetime must be an integer or noninteger greater than 0, and scale is a scaling parameter that is greater than 0, but usually 1.

METHOD GammaDist(meanLifetime, scale)
    // Needs to be greater than 0
    if meanLifetime <= 0 or scale <= 0: return error
    // Exponential distribution special case if
    // `meanLifetime` is 1 (see also Devroye 1986, p. 405)
    if meanLifetime == 1: return -ln(RNDU01ZeroOneExc()) * scale
    d = meanLifetime
    v = 0
    if meanLifetime < 1: d = d + 1
    d = d - (1.0 / 3) // NOTE: 1.0 / 3 must be a fractional number
    c = 1.0 / sqrt(9 * d)
    while true
        x = 0
        while true
           x = Normal(0, 1)
           v = c * x + 1;
           v = v * v * v
           if v > 0: break
        end
        u = RNDU01ZeroExc()
        x2 = x * x
        if u < 1 - (0.0331 * x2 * x2): break
        if ln(u) < (0.5 * x2) + (d * (1 - v + ln(v))): break
    end
    ret = d * v
    if meanLifetime < 1
       ret = ret * exp(ln(RNDU01ZeroExc()) / meanLifetime)
    end
    return ret * scale
end

Distributions based on the gamma distribution:

  • 3-parameter gamma distribution: pow(GammaDist(a, 1), 1.0 / c) * b, where c is another shape parameter.
  • 4-parameter gamma distribution: pow(GammaDist(a, 1), 1.0 / c) * b + d, where d is the minimum value.
  • Exponential distribution: GammaDist(1, 1.0 / lamda) or -ln(RNDU01ZeroOneExc()) / lamda, where lamda is the inverse scale. Usually, lamda is the probability that an independent event of a given kind will occur in a given span of time (such as in a given day or year), and the random result is the number of spans of time until that event happens. (This distribution is thus useful for modeling a Poisson process.) 1.0 / lamda is the scale (mean), which is usually the average waiting time between two independent events of the same kind.
  • Erlang distribution: GammaDist(n, 1.0 / lamda). Expresses a sum of n exponential random variables with the given lamda parameter.

Beta Distribution

In the following method, which generates a random number that follows a beta distribution, a and b are two parameters each greater than 0. The range of the beta distribution is [0, 1).

METHOD BetaDist(self, a, b, nc)
  if b==1 and a==1: return RNDU01()
  if a==1: return 1.0-pow(RNDU01(),1.0/b)
  if b==1: return pow(RNDU01(),1.0/a)
  x=GammaDist(a,1)
  return x/(x+GammaDist(b,1))
END METHOD

Note: A noncentral beta distribution is sampled by generating BetaDist(a + Poisson(nc), b), where nc is greater than 0.

Negative Binomial Distribution

A random integer that follows a negative binomial distribution expresses the number of failures that have happened after seeing a given number of successes (expressed as successes below), where the probability of a success in each case is p (where p <= 0 means never, p >= 1 means always, and p = 0.5 means an equal chance of success or failure).

METHOD NegativeBinomial(successes, p)
    // Needs to be 0 or greater
    if successes < 0: return error
    // No failures if no successes or if always succeeds
    if successes == 0 or p >= 1.0: return 0
    // Always fails (NOTE: infinity can be the maximum possible
    // integer value if NegativeBinomial is implemented to return
    // an integer)
    if p <= 0.0: return infinity
    // NOTE: If 'successes' can be an integer only,
    // omit the following three lines:
    if floor(successes) != successes
        return Poisson(GammaDist(successes, (1 - p) / p))
    end
    count = 0
    total = 0
    if successes == 1
        if p == 0.5
          while RNDINT(1) == 0: count = count + 1
           return count
        end
        // Geometric distribution special case (see Saucier 2000)
        return floor(ln(RNDU01ZeroExc()) / ln(1.0 - p))
    end
    while true
        if RNDU01OneExc() < p
            // Success
            total = total + 1
            if total >= successes
                    return count
            end
        else
            // Failure
            count = count + 1
        end
    end
END METHOD

Example: If p is 0.5 and successes is 1, the negative binomial distribution models the task "Flip a coin until you get tails, then count the number of heads."

von Mises Distribution

In the following method, which generates a random number that follows a von Mises distribution, which describes a distribution of circular angles—

  • mean is the mean angle,
  • kappa is a shape parameter, and
  • the method can return a number within π of that mean.

The algorithm below is the Best–Fisher algorithm from 1979 (as described in Devroye 1986 with errata incorporated).

METHOD VonMises(mean, kappa)
    if kappa < 0: return error
    if kappa == 0
        return RNDNUMEXCRANGE(mean-pi, mean+pi)
    end
    r = 1.0 + sqrt(4 * kappa * kappa + 1)
    rho = (r - sqrt(2 * r)) / (kappa * 2)
    s = (1 + rho * rho) / (2 * rho)
    while true
        u = RNDNUMEXCRANGE(-1, 1)
        v = RNDU01ZeroOneExc()
        z = cos(pi * u)
        w = (1 + s*z) / (s + z)
        y = kappa * (s - w)
        if y*(2 - y) - v >=0 or ln(y / v) + 1 - y >= 0
           if angle<-1: angle=-1
           if angle>1: angle=1
           // NOTE: Inverse cosine replaced here
           // with `atan2` equivalent
           angle = atan2(sqrt(1-w*w),w)
           if u < 0: angle = -angle
           return mean + angle
        end
    end
END METHOD

Stable Distribution

As more and more independent and identically distributed random variables are added together, their distribution tends to a stable distribution, which resembles a curve with a single peak, but with generally "fatter" tails than the normal distribution. The pseudocode below uses the Chambers–Mallows–Stuck algorithm. The Stable method, implemented below, takes two parameters:

  • alpha is a stability index in the interval (0, 2].
  • beta is a skewness in the interval [-1, 1]; if beta is 0, the curve is symmetric.

 

METHOD Stable(alpha, beta)
    if alpha <=0 or alpha > 2: return error
    if beta < -1 or beta > 1: return error
    halfpi = pi * 0.5
    unif=RNDNUMEXCRANGE(-halfpi, halfpi)
    while unif==-halfpi: unif=RNDNUMEXCRANGE(-halfpi, halfpi)
    // Cauchy special case
    if alpha == 1 and beta == 0: return tan(unif)
    expo=-ln(RNDU01ZeroExc())
    c=cos(unif)
    if alpha == 1
            s=sin(unif)
            return 2.0*((unif*beta+halfpi)*s/c -
                beta * ln(halfpi*expo*c/(unif*beta+halfpi)))/pi
    end
    z=-tan(alpha*halfpi)*beta
    ug=unif+atan2(-z, 1)/alpha
    cpow=pow(c, -1.0 / alpha)
    return pow(1.0+z*z, 1.0 / (2*alpha))*
        (sin(alpha*ug)*cpow)*
        pow(cos(unif-alpha*ug)/expo, (1.0 - alpha) / alpha)
END METHOD

Extended versions of the stable distribution:

  • Four-parameter stable distribution: Stable(alpha, beta) * sigma + mu, where mu is the mean and sigma is the scale. If alpha and beta are 1, the result is a Landau distribution.
  • "Type 0" stable distribution: Stable(alpha, beta) * sigma + (mu - sigma * beta * x), where x is ln(sigma)*2.0/pi if alpha is 1, and tan(pi*0.5*alpha) otherwise.

Hypergeometric Distribution

The following method generates a random integer that follows a hypergeometric distribution. When a given number of items are drawn at random without replacement from a collection of items each labeled either 1 or 0, the random integer expresses the number of items drawn this way that are labeled 1. In the method below, trials is the number of items drawn at random, ones is the number of items labeled 1 in the set, and count is the number of items labeled 1 or 0 in that set.

METHOD Hypergeometric(trials, ones, count)
    if ones < 0 or count < 0 or trials < 0 or ones > count or trials > count
            return error
    end
    if ones == 0: return 0
   successes = 0
    i = 0
    currentCount = count
    currentOnes = ones
    while i < trials and currentOnes > 0
            if RNDINTEXC(currentCount) < currentOnes
                    currentOnes = currentOnes - 1
                    successes = successes + 1
            end
            currentCount = currentCount - 1
            i = i + 1
    end
    return successes
END METHOD

Example: In a 52-card deck of Anglo-American playing cards, 12 of the cards are face cards (jacks, queens, or kings). After the deck is shuffled and seven cards are drawn, the number of face cards drawn this way follows a hypergeometric distribution where trials is 7, ones is 12, and count is 52.

Multivariate Normal (Multinormal) Distribution

The following pseudocode calculates a random point in space that follows a multivariate normal (multinormal) distribution. The method MultivariateNormal takes the following parameters:

  • A list, mu (μ), which indicates the means to add to each component of the random point. mu can be nothing, in which case each component will have a mean of zero.
  • A list of lists cov, that specifies a covariance matrix (Σ, a symmetric positive definite NxN matrix, where N is the number of components of the random point).

 

METHOD Decompose(matrix)
  numrows = size(matrix)
  if size(matrix[0])!=numrows: return error
  // Does a Cholesky decomposition of a matrix
  // assuming it's positive definite and invertible
  ret=NewList()
  for i in 0...numrows
    submat = NewList()
    for j in 0...numrows: AddItem(submat, 0)
    AddItem(ret, submat)
  end
  s1 = sqrt(matrix[0][0])
  if s1==0: return ret // For robustness
  for i in 0...numrows
    ret[0][i]=matrix[0][i]*1.0/s1
  end
  for i in 0...numrows
    sum=0.0
    for j in 0...i: sum = sum + ret[j][i]*ret[j][i]
    sq=matrix[i][i]-sum
    if sq<0: sq=0 // For robustness
    ret[i][i]=math.sqrt(sq)
  end
  for j in 0...numrows
    for i in (j + 1)...numrows
      // For robustness
      if ret[j][j]==0: ret[j][i]=0
      if ret[j][j]!=0
        sum=0
        for k in 0...j: sum = sum + ret[k][i]*ret[k][j]
        ret[j][i]=(matrix[j][i]-sum)*1.0/ret[j][j]
      end
    end
  end
  return ret
END METHOD

METHOD MultivariateNormal(mu, cov)
  mulen=size(cov)
  if mu != nothing
    mulen = size(mu)
    if mulen!=size(cov): return error
    if mulen!=size(cov[0]): return error
  end
  // NOTE: If multiple random points will
  // be generated using the same covariance
  // matrix, an implementation can consider
  // precalculating the decomposed matrix
  // in advance rather than calculating it here.
  cho=Decompose(cov)
  i=0
  ret=NewList()
  variables=NewList()
  for j in 0...mulen: AddItem(variables, Normal(0, 1))
  while i<mulen
    nv=Normal(0,1)
    sum = 0
    if mu == nothing: sum=mu[i]
    for j in 0...mulen: sum=sum+variables[j]*cho[j][i]
    AddItem(ret, sum)
    i=i+1
  end
  return ret
end

Examples:

  1. A binormal distribution (two-variable multinormal distribution) can be sampled using the following idiom: MultivariateNormal([mu1, mu2], [[s1*s1, s1*s2*rho], [rho*s1*s2, s2*s2]]), where mu1 and mu2 are the means of the two random variables, s1 and s2 are their standard deviations, and rho is a correlation coefficient greater than -1 and less than 1 (0 means no correlation).
  2. A log-multinormal distribution can be sampled by generating numbers from a multinormal distribution, then applying exp(n) to the resulting numbers, where n is each number generated this way.
  3. A Beckmann distribution can be sampled by calculating the norm of a binormal random pair (see example 1); that is, calculate sqrt(x*x+y*y), where x and y are the two numbers in the binormal pair.

Random Numbers with a Given Positive Sum

Generating N GammaDist(total, 1) numbers and dividing them by their sum will result in N random numbers that (approximately) sum to total (see a Wikipedia article). For example, if total is 1, the numbers will (approximately) sum to 1. Note that in the exceptional case that all numbers are 0, the process should repeat.

The following pseudocode shows how to generate random integers with a given positive sum. (The algorithm for this was presented in Smith and Tromble, "Sampling Uniformly from the Unit Simplex", 2004.) In the pseudocode below—

  • the method NonzeroIntegersWithSum returns n positive integers that sum to total,
  • the method IntegersWithSum returns n nonnegative integers that sum to total, and
  • Sort(list) sorts the items in list in ascending order (note that details on sort algorithms are outside the scope of this document).

 

METHOD NonzeroIntegersWithSum(n, total)
    if n <= 0 or total <=0: return error
    ls = NewList()
    ret = NewList()
    AddItem(ls, 0)
    while size(ls) < n
            c = RNDINTEXCRANGE(1, total)
            found = false
            j = 1
            while j < size(ls)
                    if ls[j] == c
                            found = true
                            break
                    end
                    j = j + 1
            end
            if found == false: AddItem(ls, c)
    end
    Sort(ls)
    AddItem(ls, total)
    for i in 1...size(ls): AddItem(ret, list[i] - list[i - 1])
    return ret
END METHOD

METHOD IntegersWithSum(n, total)
    if n <= 0 or total <=0: return error
    ret = NonzeroIntegersWithSum(n, total + n)
    for i in 0...size(ret): ret[i] = ret[i] - 1
    return ret
END METHOD

Notes:

  • Generating N random numbers with a given positive average avg is equivalent to generating N random numbers with the sum N * avg.
  • Generating N random numbers min or greater and with a given positive sum sum is equivalent to generating N random numbers with the sum sum - n * min, then adding min to each number generated this way.
  • The Dirichlet distribution, as defined in some places (e.g., Mathematica; Devroye 1986, p. 594), models n random numbers, and can be sampled by generating n+1 random gamma-distributed numbers, each with separate parameters, taking their sum, and dividing the first n numbers by that sum.

Multinomial Distribution

The multinomial distribution models the number of times each of several mutually exclusive events happens among a given number of trials, where each event can have a separate probability of happening. The pseudocode below is of a method that takes two parameters: trials, which is the number of trials, and weights, which are the relative probabilities of each event. The method tallies the events as they happen and returns a list (with the same size as weights) containing the number of successes for each event.

METHOD Multinomial(trials, weights)
    if trials < 0: return error
    // create a list of successes
    list = NewList()
    for i in 0...size(weights): AddItem(list, 0)
    for i in 0...trials
        // Choose an index
        index = WeightedChoice(weights)
        // Tally the event at the chosen index
        list[index] = list[index] + 1
    end
    return list
END METHOD

Gaussian and Other Copulas

A copula is a distribution describing the correlation (dependence) between random numbers.

One example is a Gaussian copula; this copula is sampled by sampling from a multinormal distribution, then converting the resulting numbers to uniformly-distributed, but correlated, numbers. In the following pseudocode, which implements a Gaussian copula:

  • The parameter covar is the covariance matrix for the multinormal distribution.
  • erf(v) is the error function of the variable v (see the appendix).

 

METHOD GaussianCopula(covar)
   mvn=MultivariateNormal(nothing, covar)
   for i in 0...size(covar)
      // Apply the normal distribution's CDF
      // to get uniform variables
      mvn[i] = (erf(mvn[i]/(sqrt(2)*sqrt(covar[i][i])))+1)*0.5
   end
   return mvn
END METHOD

Each of the resulting uniform numbers will be in the interval [0, 1], and each one can be further transformed to any other probability distribution (which is called a marginal distribution here) by one of the methods given in "Random Numbers from an Arbitrary Distribution". (See also Cario and Nelson 1997.)

Examples:

  1. To generate two correlated uniform variables with a Gaussian copula, generate GaussianCopula([[1, rho], [rho, 1]]), where rho is the Pearson correlation coefficient, in the interval [-1, 1]. (Note that rank correlation parameters, which can be converted to rho, can better describe the correlation than rho itself. For example, for a two-variable Gaussian copula, the Spearman coefficient srho can be converted to rho by rho = sin(srho * pi / 6) * 2. Rank correlation parameters are not further discussed in this document.)
  2. The following example generates two random numbers that follow a Gaussian copula with exponential marginals (rho is the Pearson correlation coefficient, and rate1 and rate2 are the rates of the two exponential marginals).

    METHOD CorrelatedExpo(rho, rate1, rate2)
       copula = GaussianCopula([[1, rho], [rho, 1]])
       // Transform to exponentials using that
       // distribution's inverse CDF
       return [-ln(copula[0]) / rate1, -ln(copula[1]) / rate2]
    END METHOD

Other kinds of copulas describe different kinds of correlation between random numbers. Examples of other copulas are—

  • the Fréchet–Hoeffding upper bound copula [x, x, ..., x] (e.g., [x, x]), where x = RNDU01(),
  • the Fréchet–Hoeffding lower bound copula [x, 1.0 - x] where x = RNDU01(),
  • the product copula, where each number is a separately generated RNDU01() (indicating no correlation between the numbers), and
  • the Archimedean copulas, described by M. Hofert and M. Mächler (2011)(14).

Other Non-Uniform Distributions

Most commonly used:

  • Cauchy (Lorentz) distribution: scale * tan(pi * (RNDU01OneExc()-0.5)) + mu, where scale is the scale and mu is the location of the distribution's curve peak (mode). This distribution is similar to the normal distribution, but with "fatter" tails.
  • Chi-squared distribution: GammaDist(df * 0.5 + Poisson(sms * 0.5), 2), where df is the number of degrees of freedom and sms is the sum of mean squares (where sms other than 0 indicates a noncentral distribution).
  • Extreme value distribution: a - ln(-ln(RNDU01ZeroOneExc())) * b, where b is the scale and a is the location of the distribution's curve peak (mode). This expresses a distribution of maximum values.
  • Geometric distribution: NegativeBinomial(1, p), where p has the same meaning as in the negative binomial distribution. As used here, this is the number of failures that have happened before a success happens. (Saucier 2000, p. 44, also mentions an alternative definition that includes the success.)
  • Gumbel distribution: a + ln(-ln(RNDU01ZeroOneExc())) * b, where b is the scale and a is the location of the distribution's curve peak (mode). This expresses a distribution of minimum values.
  • Inverse gamma distribution: b / GammaDist(a, 1), where a and b have the same meaning as in the gamma distribution. Alternatively, 1.0 / (pow(GammaDist(a, 1), 1.0 / c) / b + d), where c and d are shape and location parameters, respectively.
  • Laplace (double exponential) distribution: (ln(RNDU01ZeroExc()) - ln(RNDU01ZeroExc())) * beta + mu, where beta is the scale and mu is the mean.
  • Logarithmic distribution: min + (max - min) * RNDU01OneExc() * RNDU01OneExc(), where min is the minimum value and max is the maximum value (Saucier 2000, p. 26). In this distribution, numbers closer to min are exponentially more likely than numbers closer to max.
  • Logarithmic normal distribution: exp(Normal(mu, sigma)), where mu and sigma have the same meaning as in the normal distribution.
  • Pareto distribution: pow(RNDU01ZeroOneExc(), -1.0 / alpha) * minimum, where alpha is the shape and minimum is the minimum.
  • Rayleigh distribution: a * sqrt(-2 * ln(RNDU01ZeroExc())), where a is the scale and is greater than 0. If a follows a logarithmic normal distribution, the result is a Suzuki distribution.
  • Student's t-distribution: Normal(cent, 1) / sqrt(GammaDist(df * 0.5, 2 / df)), where df is the number of degrees of freedom, and cent is the mean of the normally-distributed random number. A cent other than 0 indicates a noncentral distribution.
  • Triangular distribution: ContinuousWeightedChoice([startpt, midpt, endpt], [0, 1, 0]). The distribution starts at startpt, peaks at midpt, and ends at endpt.
  • Weibull distribution: b * pow(-ln(RNDU01ZeroExc()),1.0 / a) + loc, where a is the shape, b is the scale loc is the location, and a and b are greater than 0.

Miscellaneous:

  • Arcsine distribution: min + (max - min) * BetaDist(0.5, 0.5), where min is the minimum value and max is the maximum value (Saucier 2000, p. 14).
  • Beta binomial distribution: Binomial(trials, BetaDist(a, b)), where a and b are the two parameters of the beta distribution, and trials is a parameter of the binomial distribution.
  • Beta-PERT distribution: startpt + size * BetaDist(1.0 + (midpt - startpt) * shape / size, 1.0 + (endpt - midpt) * shape / size). The distribution starts at startpt, peaks at midpt, and ends at endpt, size is endpt - startpt, and shape is a shape parameter that's 0 or greater, but usually 4. If the mean (mean) is known rather than the peak, midpt = 3 * mean / 2 - (startpt + endpt) / 4.
  • Beta prime distribution: pow(GammaDist(a, 1), 1.0 / alpha) * scale / pow(GammaDist(b, 1), 1.0 / alpha), where a, b, and alpha are shape parameters and scale is the scale. If a is 1, the result is a Singh–Maddala distribution; if b is 1, a Dagum distribution; if a and b are both 1, a logarithmic logistic distribution.
  • Beta negative binomial distribution: NegativeBinomial(successes, BetaDist(a, b)), where a and b are the two parameters of the beta distribution, and successes is a parameter of the negative binomial distribution. If successes is 1, the result is a Waring–Yule distribution.
  • Birnbaum–Saunders distribution: pow(sqrt(4+x*x)+x,2)/(4.0*lamda), where x = Normal(0,gamma), gamma is a shape parameter, and lamda is a scale parameter.
  • Chi distribution: sqrt(GammaDist(df * 0.5, 2)), where df is the number of degrees of freedom.
  • Cosine distribution: min + (max - min) * atan2(x, sqrt(1 - x * x)) / pi, where x = RNDNUMRANGE(-1, 1) and min is the minimum value and max is the maximum value (Saucier 2000, p. 17; inverse sine replaced with atan2 equivalent).
  • Double logarithmic distribution: min + (max - min) * (0.5 + (RNDINT(1) * 2 - 1) * 0.5 * RNDU01OneExc() * RNDU01OneExc()), where min is the minimum value and max is the maximum value (see also Saucier 2000, p. 15, which shows the wrong X axes).
  • Fréchet distribution: b*pow(-ln(RNDU01ZeroExc()),-1.0/a) + loc, where a is the shape, b is the scale, and loc is the location of the distribution's curve peak (mode). This expresses a distribution of maximum values.
  • Generalized extreme value (Fisher–Tippett) distribution: a - (pow(-ln(RNDU01ZeroOneExc()), -c) - 1) * b / c if c != 0, or a - ln(-ln(RNDU01ZeroOneExc())) * b otherwise, where b is the scale, a is the location of the distribution's curve peak (mode), and c is a shape parameter. This expresses a distribution of maximum values.
  • Generalized Tukey lambda distribution: (s1 * (pow(x, lamda1)-1.0)/lamda1 - s2 * (pow(1.0-x, lamda2)-1.0)/lamda2) + loc, where x is RNDU01(), lamda1 and lamda2 are shape parameters, s1 and s2 are scale parameters, and loc is a location parameter.
  • Half-normal distribution. Parameterizations include:
    • Mathematica: abs(Normal(0, sqrt(pi * 0.5) / invscale))), where invscale is a parameter of the half-normal distribution.
    • MATLAB: abs(Normal(mu, sigma))), where mu and sigma are the same as in the normal distribution.
  • Inverse chi-squared distribution: df * scale / (GammaDist(df * 0.5, 2)), where df is the number of degrees of freedom and scale is the scale, usually 1.0 / df.
  • Inverse Gaussian distribution (Wald distribution): Generate n = mu + (mu*mu*y/(2*lamda)) - mu * sqrt(4 * mu * lamda * y + mu * mu * y * y) / (2 * lamda), where y = pow(Normal(0, 1), 2), then return n if RNDU01OneExc() <= mu / (mu + n), or mu * mu / n otherwise. mu is the mean and lamda is the scale; both parameters are greater than 0. Based on method published in Devroye 1986.
  • Kumaraswamy distribution: min + (max - min) * pow(1-pow(RNDU01ZeroExc(),1.0/b),1.0/a), where a and b are shape parameters, min is the minimum value, and max is the maximum value.
  • Lévy distribution: sigma * 0.5 / GammaDist(0.5, 1) + mu, where mu is the location and sigma is the dispersion.
  • Logarithmic series distribution: floor(1.0 + ln(RNDU01ZeroExc()) / ln(1.0 - pow(1.0 - param, RNDU01ZeroOneExc()))), where param is a number greater than 0 and less than 1. Based on method described in Devroye 1986.
  • Logistic distribution: (ln(x)-ln(1.0 - x)) * scale + mean, where x is RNDU01ZeroOneExc() and mean and scale are the mean and the scale, respectively.
  • Maxwell distribution: scale * sqrt(GammaDist(1.5, 2)), where scale is the scale.
  • Parabolic distribution: min + (max - min) * BetaDist(2, 2), where min is the minimum value and max is the maximum value (Saucier 2000, p. 30).
  • Pascal distribution: NegativeBinomial(successes, p) + successes, where successes and p have the same meaning as in the negative binomial distribution, except successes is always an integer.
  • Pearson VI distribution: GammaDist(v, 1) / (GammaDist(w, 1)), where v and w are shape parameters greater than 0 (Saucier 2000, p. 33; there, an additional b parameter is defined, but that parameter is canceled out in the source code).
  • Power distribution: pow(RNDU01ZeroOneExc(), 1.0 / alpha) / b, where alpha is the shape and b is the domain. Nominally in the interval (0, 1).
  • Power law distribution: pow(pow(mn,n+1) + (pow(mx,n+1) - pow(mn,n+1)) * RNDU01(), 1.0 / (n+1)), where n is the exponent, mn is the minimum, and mx is the maximum. Reference.
  • Skellam distribution: Poisson(mean1) - Poisson(mean2), where mean1 and mean2 are the means of the two Poisson variables.
  • Skewed normal distribution: Normal(0, x) + mu + alpha * abs(Normal(0, x)), where x is sigma / sqrt(alpha * alpha + 1.0), mu and sigma have the same meaning as in the normal distribution, and alpha is a shape parameter.
  • Snedecor's (Fisher's) F-distribution: GammaDist(m * 0.5, n) / (GammaDist(n * 0.5 + Poisson(sms * 0.5)) * m, 1), where m and n are the numbers of degrees of freedom of two random numbers with a chi-squared distribution, and if sms is other than 0, one of those distributions is noncentral with sum of mean squares equal to sms.
  • Tukey lambda distribution: (pow(x, lamda)-pow(1.0-x,lamda))/lamda, where x is RNDU01() and lamda is a shape parameter (if 0, the result is a logistic distribution).
  • Zeta distribution: Generate n = floor(pow(RNDU01ZeroOneExc(), -1.0 / r)), and if d / pow(2, r) < (d - 1) * RNDU01OneExc() * n / (pow(2, r) - 1.0), where d = pow((1.0 / n) + 1, r), repeat this process. The parameter r is greater than 0. Based on method described in Devroye 1986. A zeta distribution truncated by rejecting random values greater than some positive integer is called a Zipf distribution or Estoup distribution. (Note that Devroye uses "Zipf distribution" to refer to the untruncated zeta distribution.)

The Python sample code also contains implementations of the power normal distribution, the power lognormal distribution, the negative multinomial distribution, the multivariate t-distribution, the multivariate t-copula, and the multivariate Poisson distribution.

Geometric Sampling

This section contains various geometric sampling techniques.

Random Points Inside a Simplex

The following pseudocode generates, uniformly at random, a point inside an n dimensional simplex (simplest convex figure, such as a line segment, triangle, or tetrahedron). It takes an array points, a list consisting of the n plus one vertices of the simplex, all of a single dimension n or greater.

METHOD RandomPointInSimplex(points):
   ret=NewList()
   if size(points) > size(points[0])+1: return error
   if size(points)==1 // Return a copy of the point
     for i in 0...size(points[0]): AddItem(ret,points[0][i])
     return ret
   end
   gammas=NewList()
   // Sample from a Dirichlet distribution
   simplexDims=size(points)-1
   for i in 0...size(points): AddItem(gammas, -ln(RNDU01ZeroOneExc()))
   tsum=0
   for i in 0...size(gammas): tsum = tsum + gammas[i]
   tot = 0
   for i in 0...size(gammas) - 1
       gammas[i] = gammas[i] / tsum
       tot = tot + gammas[i]
   end
   tot = 1.0 - tot
   for i in 0...size(points[0]): AddItem(ret, points[0][i]*tot)
   for i in 1...size(points)
      for j in 0...size(points[0])
         ret[j]=ret[j]+points[i][j]*gammas[i-1]
      end
   end
   return ret
END METHOD

Random Points on the Surface of a Hypersphere

To generate, uniformly at random, an N-dimensional point on the surface of an N-dimensional hypersphere of radius R, generate N Normal(0, 1) random numbers, then multiply them by R / X, where X is those numbers' norm (if X is 0, the process should repeat). A hypersphere's surface is formed by all points lying R units away from a common point in N-dimensional space. Based on a technique described in MathWorld.

This problem is equivalent to generating, uniformly at random, a unit vector (vector with length 1) in N-dimensional space.

Random Points Inside a Ball or Shell

To generate, uniformly at random, an N-dimensional point inside an N-dimensional ball of radius R, either—

  • generate N Normal(0, 1) random numbers, generate X = sqrt( S - ln(RNDU01ZeroExc())), where S is the sum of squares of the random numbers, and multiply each random number by R / X (if X is 0, the process should repeat), or
  • generate N RNDNUMRANGE(-R, R) random numbers(15) until their norm is R or less,

although the former method "may ... be slower" "in practice", according to a MathWorld article, which was the inspiration for the two methods given here.

To generate, uniformly at random, a point inside an N-dimensional spherical shell (a hollow ball) with inner radius A and outer radius B (where A is less than B), either—

  • generate, uniformly at random, a point for a ball of radius B until the norm of the numbers making up that point is A or greater;
  • for 2 dimensions, generate, uniformly at random, a point on the surface of a circle with radius equal to sqrt(RNDNUMRANGE(0, B * B - A * A) + A * A) (Dupree and Fraley 2004); or
  • for 3 dimensions, generate, uniformly at random, a point on the surface of a sphere with radius equal to pow(RNDNUMRANGE(0, pow(B, 3) - pow(A, 3)) + pow(A, 3), 1.0 / 3.0) (Dupree and Fraley 2004).

Random Latitude and Longitude

To generate, uniformly at random, a point on the surface of a sphere in the form of a latitude and longitude (in radians with west and south coordinates negative)—

  • generate the longitude RNDNUMEXCRANGE(-pi, pi), where the longitude ranges from -π to π, and
  • generate the latitude atan2(sqrt(1 - x * x), x) - pi / 2, where—
    • x = RNDNUMRANGE(-1, 1) and the latitude ranges from -π/2 to π/2 (the range includes the poles, which have many equivalent forms), or
    • x = 2 * RNDU01ZeroOneExc() - 1 and the latitude ranges from -π/2 to π/2 (the range excludes the poles).

Reference: "Sphere Point Picking" in MathWorld (replacing inverse cosine with atan2 equivalent).

Conclusion

This page discussed many ways applications can extract random numbers from random number generators.

I acknowledge the commenters to the CodeProject version of this page, including George Swan, who referred me to the reservoir sampling method.

Notes

(1) An RNG meeting this definition can—

  • seek to generate random numbers that are at least cost-prohibitive (but not necessarily impossible) to predict,
  • merely seek to generate number sequences likely to pass statistical tests of randomness,
  • be initialized automatically before use,
  • be initialized with an application-specified "seed",
  • use a deterministic algorithm for random number generation,
  • rely, at least primarily, on one or more nondeterministic sources for random number generation (including by extracting uniformly distributed bits from two or more such sources), or
  • have two or more of the foregoing properties.

If the software and/or hardware uses a nonuniform distribution, but otherwise meets this definition, it can be converted to use a uniform distribution, at least in theory, using unbiasing, deskewing, or randomness extraction, which are outside the scope of this document.

(2) For an exercise solved by this method, see A. Koenig and B. E. Moo, Accelerated C++, 2000; see also a blog post by Johnny Chan. In addition, M. O'Neill discusses various methods, both biased and unbiased, for generating random integers in a range with an RNG in a blog post from July 2018.

Note that if MODULUS is a power of 2 (for example, 256 or 232), the RNDINT implementation given in the pseudocode may leave unused bits (for example, when truncating a random number to wordBits bits or in the special cases at the start of the method). How a more sophisticated implementation may save those bits for later reuse is beyond this page's scope.

(3) This number format describes B-bit signed integers with minimum value -2B-1 and maximum value 2B-1 - 1, where B is a positive even number of bits; examples include Java's short, int, and long, with 16, 32, and 64 bits, respectively. A signed integer is an integer that can be positive, zero, or negative. In two's-complement form, nonnegative numbers have the highest (most significant) bit set to zero, and negative numbers have that bit (and all bits beyond) set to one, and a negative number is stored in such form by swapping the bits of a number equal to that number's absolute value minus 1.

(4) RNDINTEXC is not given as the core random generation method because it's harder to fill integers in popular integer formats with random bits with this method.

(5) In situations where loops are not possible, such as within an SQL query, the idiom min(floor(RNDU01OneExc() * maxExclusive, maxExclusive - 1)) returns an integer in the interval [0, maxExclusive); however, such an idiom can have a slight, but for most purposes negligible, bias toward maxExclusive - 1.

(6) The following illustrates some of the differences in RNGs between SQL dialects. MySQL and T-SQL both have a RAND() akin to RNDU01OneExc(); however, T-SQL's RAND() is constant per expression, rather than per row. PL/SQL (Oracle) has DBMS_RANDOM.VALUE akin to either RNDU01OneExc() or RNDNUMEXCRANGE. PostgreSQL has RANDOM(). SQLite sometimes has RANDOM() which returns a random integer in the interval [-263, 263). Most likely, none of these were designed for information security.

(7) It suffices to say here that in general, whenever a deterministic RNG is otherwise called for, such an RNG is good enough for shuffling a 52-item list if its period is 2226 or greater. (The period is the maximum number of values in a generated sequence for a deterministic RNG before that sequence repeats.)

(8) A convex polygon can be trivially decomposed into triangles that have one vertex in common and each have two other adjacent vertices of the original polygon. Triangulation of other polygons is nontrivial and outside the scope of this document.

(9) "Jitter", as used in this step, follows a distribution formally called a kernel, of which the normal distribution is one example. Bandwidth should be as low or as high as allows the estimated distribution to fit the data and remain smooth. A more complex kind of "jitter" (for multi-component data points) consists of a point generated from a multinormal distribution with all the means equal to 0 and a covariance matrix that, in this context, serves as a bandwidth matrix. "Jitter" and bandwidth are not further discussed in this document.

(10) That article also mentions a critical-hit distribution, which is actually a mixture of two distributions: one roll of dice and the sum of two rolls of dice.

(11) More formally—

  • the PDF is the derivative (instantaneous rate of change) of the distribution's CDF (that is, PDF(x) = CDF′(x)), and
  • the CDF is also defined as the integral of the PDF,

provided the PDF's values are all 0 or greater and the area under the PDF's curve is 1.

(12) The "Dice" section used the following sources:

  • Red Blob Games, "Probability and Games: Damage Rolls" was the main source for the dice-roll distribution. The method random(N) in that document corresponds to RNDINTEXC(N) in this document.
  • The MathWorld article "Dice" provided the mean of the dice roll distribution.
  • S. Eger, "Stirling's approximation for central extended binomial coefficients", 2014, helped suggest the variance of the dice roll distribution.

(13) The method that formerly appeared here is the Box-Muller transformation: mu + radius * cos(angle) and mu + radius * sin(angle), where angle = 2 * pi * RNDU01OneExc() and radius = sqrt(-2 * ln(RNDU01ZeroExc())) * sigma, are two independent normally-distributed random numbers. A method of generating approximate standard normal random numbers, which consists of summing twelve RNDU01OneExc() numbers and subtracting by 6 (see also "Irwin–Hall distribution" on Wikipedia), results in values not less than -6 or greater than 6; on the other hand, in a standard normal distribution, results less than -6 or greater than 6 will occur only with a generally negligible probability.

(14) Hofert, M., and Maechler, M. "Nested Archimedean Copulas Meet R: The nacopula Package". Journal of Statistical Software 39(9), 2011, pp. 1-20.

(15) The N numbers generated this way will form a point inside an N-dimensional hypercube with length 2 * R in each dimension and centered at the origin of space.

Appendix

 

Implementation of erf

The pseudocode below shows how the error function erf can be implemented, in case the programming language used doesn't include a built-in version of erf (such as JavaScript at the time of this writing). In the pseudocode, EPSILON is a very small number to end the iterative calculation.

METHOD erf(v)
    if v==0: return 0
    if v<0: return -erf(-v)
    if v==infinity: return 1
    // NOTE: For Java `double`, the following
    // line can be added:
    // if v>=6: return 1
    i=1
    ret=0
    zp=-(v*v)
    zval=1.0
    den=1.0
    while i < 100
        r=v*zval/den
        den=den+2
        ret=ret+r
        // NOTE: EPSILON can be pow(10,14),
        // for example.
        if abs(r)<EPSILON: break
        if i==1: zval=zp
        else: zval = zval*zp/i
        i = i + 1
    end
    return ret*2/sqrt(pi)
END METHOD

Mean and Variance Calculation

The following method calculates the mean and the bias-corrected sample variance of a list of real numbers. It returns a two-item list containing the mean and that kind of variance in that order. It uses the Welford method presented by J. D. Cook.

METHOD MeanAndVariance(list)
    if size(list)==0: return [0, 0]
    if size(list)==1: return [list[0], 0]
    xm=list[0]
    xs=0
    i=1
    while i < len(list)
        c = list[i]
        i = i + 1
        cxm = (c - xm)
        xm = xm + cxm *1.0/ i
        xs = xs + cxm * (c - xm)
    end
    return [xm, xs*1.0/(size(list)-1)]
END

License

This article, along with any associated source code and files, is licensed under A Public Domain dedication

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GeneralMessage Closed Pin
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QuestionPython Pin
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QuestionTo the author Pin
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memberMember 117206815-Sep-17 13:19 
Suggestiona suggestion or two Pin
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GeneralRe: a suggestion or two Pin
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SuggestionCorrection Pin
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GeneralRe: Correction Pin
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QuestionPeter Occil article "Random Number Generation" Pin
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AnswerRe: Peter Occil article "Random Number Generation" Pin
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QuestionChoosing Several Items from a Dataset Pin
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QuestionShuffle Pin
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Suggestionbetter if there are more explanation about the algorithm ,for example ,the box-muller alg Pin
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