Generating Random Numbers

Random numbers are important resources for scientific applications, education, game development and visualization.

The standard RTL function random generates random numbers that fulfill a uniform distribution. Uniformly distributed random numbers are not useful for every application. In order to create random numbers of other distributions special algorithms are necessary.

Normal (Gaussian) Distribution

One of the more common algorithms to produce normally distributed random numbers from uniformly distributed random numbers is the Box-Müller approach. The following function calculates Gaussian-distributed random numbers:

 function rnorm (mean, sd: real): real;
 {Calculates Gaussian random numbers according to the Box-Müller approach}
  var
   u1, u2: real;
 begin
   u1 := random;
   u2 := random;
   rnorm := mean * abs(1 + sqrt(-2 * (ln(u1))) * cos(2 * pi * u2) * sd);
  end;

The same algorithm is used by the randg randg function from the RTL math unit:

function randg(mean,stddev: float): float;

Exponential Distribution

An exponential distribution occurs frequently in real-world problems. A classical example is the distribution of waiting times between independent Poisson-random events, e.g. the radioactive decay of nuclei [Press et al. 1989].

The following function delivers a single real random number out of an exponential distribution. Rate is the inverse of the mean and the constant RESOLUTION determines the granularity of generated random numbers.

function randomExp(a, rate: real): real;
const
  RESOLUTION = 1000;
var
  unif: real;
begin
  if rate = 0 then
    randomExp := NaN
  else
  begin
    repeat
      unif := random(RESOLUTION) / RESOLUTION;
    until unif <> 0;
    randomExp := a - rate * ln(unif);
  end;
end;

Gamma Distribution

The gamma distribution is a two-parameter family of continuous random distributions. It is a generalization of both the exponential distribution and the Erlang distribution. Possible applications of the gamma distribution include modelling and simulation of waiting lines, or queues, and actuarial science.

The following function delivers a single real random number out of a gamma distribution. The shape of the distribution is defined by the parameters a, b and c. The function makes use of the function randomExp as defined above.

  function randomGamma(a, b, c: real): real;
  const
    RESOLUTION = 1000;
    T = 4.5;
    D = 1 + ln(T);
  var
    unif: real;
    A2, B2, C2, Q, p, y: real;
    p1, p2, v, w, z: real;
    found: boolean;
  begin
    A2 := 1 / sqrt(2 * c - 1);
    B2 := c - ln(4);
    Q := c + 1 / A2;
    C2 := 1 + c / exp(1);
    found := false;
    if c < 1 then
    begin
      repeat
        repeat
          unif := random(RESOLUTION) / RESOLUTION;
        until unif > 0;
        p := C2 * unif;
        if p > 1 then
        begin
          y := -ln((C2 - p) / c);
          if unif <= power(y, c - 1) then
          begin
            randomGamma := a + b * y;
            found := true;
          end;
        end
        else
        begin
          y := power(p, 1 / c);
          if unif <= exp(-y) then begin
            randomGamma := a + b * y;
            found := true;
          end;
        end;
      until found;
    end
    else if c = 1 then
    { Gamma distribution becomes exponential distribution, if c = 1 }
    begin
      randomGamma := randomExp(a, 1/b);
    end
    else
    begin
      repeat
        repeat
          p1 := random(RESOLUTION) / RESOLUTION;
        until p1 > 0;
        repeat
          p2 := random(RESOLUTION) / RESOLUTION;
        until p2 > 0;
        v := A2 * ln(p1 / (1 - p1));
        y := c * exp(v);
        z := p1 * p1 * p2;
        w := B2 + Q * v - y;
        if (w + D - T * z >= 0) or (w >= ln(z)) then
        begin
          randomGamma := a + b * y;
          found := true;
        end;
      until found;
    end;
  end;

Erlang Distribution

The Erlang distribution is a two parameter family of continuous probability distributions. It is a generalization of the exponential distribution and a special case of the gamma distribution, where c is an integer. The Erlang distribution has been first described by Agner Krarup Erlang in order to model the time interval between telephone calls. It is used for queuing theory and for simulating waiting lines.

  function randomErlang(mean: real; k: integer): real;
  const
    RESOLUTION = 1000;
  var
    i: integer;
    unif, prod: real;
  begin
    if (mean <= 0) or (k < 1) then
      randomErlang := NaN
    else
    begin
      prod := 1;
      for i := 1 to k do
      begin
        repeat
          unif := random(RESOLUTION) / RESOLUTION;
        until unif <> 0;
        prod := prod * unif;
      end;
      randomErlang := -mean * ln(prod);
    end;
  end;

Chi Squared Distribution

The chi squared distribution is a continuous distribution of random numbers with df degrees of freedom. It is the distribution of a sum of the squares of df independent standard normal random variables. The chi squared distribution has numerous applications in inferential statistics, e.g. in estimating variances and for chi-squared tests. It is a special gamma distribution with c = df/ 2 and b = 2. Therefore the following function depends from the function randomGamma.

function randomChisq(df: integer): real;
begin
  if df < 1 then randomChisq := NaN
  else
  randomChisq := randomGamma(0, 2, 0.5 * df);
end;

See also

• Dev random
• Functions for descriptive statistics

References

1. G. E. P. Box and Mervin E. Muller, A Note on the Generation of Random Normal Deviates, The Annals of Mathematical Statistics (1958), Vol. 29, No. 2 pp. 610–611
2. Dietrich, J. W. (2002). Der Hypophysen-Schilddrüsen-Regelkreis. Berlin, Germany: Logos-Verlag Berlin. ISBN 978-3-89722-850-4. OCLC 50451543.
3. Press, W. H., B. P. Flannery, S. A. Teukolsky, W. T. Vetterling (1989). Numerical Recipes in Pascal. The Art of Scientific Computing, Cambridge University Press, ISBN 0-521-37516-9.
4. Richard Saucier, Computer Generation of Statistical Distributions, ARL-TR-2168, US Army Research Laboratory, Aberdeen Proving Ground, MD, 21005-5068, March 2000.
5. R.U. Seydel, Generating Random Numbers with Specified Distributions. In: Tools for Computational Finance, Universitext, DOI 10.1007/978-1-4471-2993-6_2, © Springer-Verlag London Limited 2012
6. Christian Walck, Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists, Internal Report SUF–PFY/96–01, University of Stockholm 2007