Some excerpts or summaries about this subject are presented from

Ravindran et al. [1987]

Ecker et al. [1988]

   Computer programs are available (to the IST Unix Alfa users) for verification of some of the proposed algorithms.  See also "Monte Carlo" in the calculations page.

Uniform random numbers
   [Hillier et al., 1995] The random numbers initially generated by a computer usually are random integers.  However, if desired, each of these numbers can immediately be converted to a (** continuous) uniform random number as follows.
   For a given random integer number in the range 0 to n, dividing this number by n yields (approximately) a uniform random number.  If n is small, this approximation should be improved by adding ½ to the random integer number and then dividing by n + 1 instead (** recommended). 

Generating random variables (general case, nonuniform random numbers)
   [Wagner, 1972, p 901] The following (** "inverse transform method") is the simplest and most fundamental technique for simulating random draws from an arbitrary single-variable probability distribution.  Let the distribution function for the random variable be denoted by

Now, 0 ≤ F(x) ≤ 1, and suppose F(x) is continuous and strictly increasing.  Then, given a value r, where 0 ≤ r ≤ 1, there is a unique value for x such that F(x) = r.  This value is

(** This simple yet elegant sampling rule was first suggested by von Neumann in a letter to Ulam in 1947 [Los Alamos Science, p. 135, June 1987]. It is sometimes called the "Golden Rule for Sampling".)

The technique is to generate a sequence of uniform random decimal numbers r_i, i = 1..n, and from these, determine the associated values of x_i = F⁻¹(r_i).
   (** The following correction may be necessary.) Suppose the random numbers r_i are in the range .000 to .999 (three digits).  Then, .0005 can be added to the value of each of the r_i.  This would obviously be convenient for the Gauss distribution.  (See the "lens".)
   For a discrete variable, x = 0..n (** not forcibly from 0), with F(x) = ∑_i=0^x p_i, then the inverse function can be written as

where we let F(−1) = 0.  See an example in Excel.

Conventions:
†: indicates a questionable statement;
**: a note.

References:
• Ecker, Joseph G., Michael Kupferschmid, 1988, "Introduction to Operations Research", John Wiley & Sons, New York, NY, USA
• Hillier, Frederick S., Gerald J. Lieberman, 1995, "Introduction to Operations Research", 6.th ed., McGraw-Hill, New York, NY, USA
• Ravindran, A., Don T. Phillips, James J. Solberg, 1987, "Operations Research: principles and practice", 2.nd ed., John Wiley & Sons, New York, NY, USA
• Wagner, Harvey M., 1972, "Principles of Operations Research (with applications to managerial decisions)", Prentice-Hall International, Inc., London, UK

Created: 12-Oct-2003 — Last update: 13-Oct-2003