• Uniform [a,b]

  If the objective is to generate random numbers in a range [a,b] (i.e., between a and b). The uniform process generator is:

  random-variate = a+ (b-a) * rnd

  Note that when rnd returns 0, random-variate is equal to a.
  Note that when rnd returns 1, random-variate is equal to b.
  Note that when rnd returns a number between 0 and 1, the distance between a and b (i.e., the term b minus a) is randomly divided by the value of rnd and then added to a. In effect we are using the random value generated by rnd as a random proportion (percentage) to choose a number between a and b.

• Normal (x,s) (mean=x, standard deviation=s)

  The simple way is to first generate n(0,1) (i.e., mean zero, and standard deviation1). One simple way to do this is based on an old IBM Scientific Subroutine Package method. The method is:

  1. sum up 12 uniform numbers between 0 and 1
  2. subtract the sum from 6
  3. the result is approximately n(0,1)

  We will use this result as a z score (or t score): Given that we know:

  z =	mean - observation std. dev.

  We can rearrange the terms so that:

  observation =

  mean - ( std. dev. * z )

  This is the process generator for n(x,s). To obtain a random variate from a distribution which is n(x,s), x (the mean) is known and s (the standard deviation) is known. The steps are:

  1. First generate a random n(0,1) (call this z) by:

     z=0
     for i=1 to 12
       z=z+rnd
     next
     z=6-z

  2. then calculate:

     random-variate= x - s * z

• Discrete Distributions

  Often we are attempting to create a process that follows some discrete probability distribution. We thus have events and probabilities that each event occurs, like:

  Event	Probability
  A	0.10
  B	0.35
  C	0.25
  D	0.15
  E	0.15
  sum	1.00

  First, we create a "cumulative probability distribution" from the probability distribution by accumulating the probabilities of each event:

  Event	Probability	Cumulative Probability
  A	0.10	0.10
  B	0.35	0.45
  C	0.25	0.70
  D	0.15	0.85
  E	0.15	1.00
  sum	1.00

  We can use this cumulative distribution to map from our uniform (0,1) random variates to the cumulative distribution.

  Event	Probability	Cumulative Probability	Event associated with random values:
  A	0.10	0.10	0.00 to 0.10
  B	0.35	0.45	0.10 to 0.45
  C	0.25	0.70	0.45 to 0.70
  D	0.15	0.85	0.70 to 0.85
  E	0.15	1.00	0.85 to 1.00
  sum	1.00

  Note 10% of the rnd function's values are between 0.00 and 0.10 (this can be associated with the event A)
  Note 35% of the rnd function's values are between 0.10 and 0.45 (this can be associated with the event B)
  Note 25% of the rnd function's values are between 0.45 and 0.70 (this can be associated with the event C)
  Note 15% of the rnd function's values are between 0.70 and 0.85 (this can be associated with the event D)
  Note 15% of the rnd function's values are between 0.85 and 1.00 (this can be associated with the event E)
  

  Thus, we create random occurences in the same proportion as in the original distribution.

Here are some examples:

Pick an example 1,000 Uniform Random [0,100] 1,000 Normal (0,1) mean = 0 standard deviation = 1 1,000 dice rolls (sum of two six sided) 1,000 from a Discrete Distribution (example from above)

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