by Felisa J. Vázquez-Abad and Yanick Champoux

# Generation of Random Variables

Cross Referencing:
- Topics Index
- SimSpiders Main Page

## General Discrete Distribution

The Inverse Function Method: Suppose that k0=0. The Inverse Function Method then requires setting N(U) = min{k : F(k+1) > U}.

• Generate U~U(0,1)
• F(0) = p0, k = 0.
• While (F(k) < U)
k = k+1, F(k) = F(k-1) + pk
• Set N = k

Notice that the average number of iterations will always be E[N] + 1 when the above algorithm is used. This is so because the number of times that the algorithm passes through the while operation is precisely N+1, where N is the final value.

Accelerating Techniques: Buckets The Method of Buckets combines the Composition and the Inverse Function methods for generating random variables and works well especially when generating empirical discrete distributions with a large number of values. See Bratley, Fox and Sxrhage (**reference). A total of M buckets are defined by the subintervals [m/M, (m+1)/M) for m=0, ..., M-1. The m-th bucket is composed of all those indices k such that F(k) lies within [m/M, (m+1)/M) plus the last element. Call b(m) the first element of the m-th bucket.

 The figure shows the bucket B2 containing elements {1,2,3}. when we generate a random variable U, instead of searching as in the previous algorithm, we now narrow the search by looking for the bucket and then finiding the value within the bucket. Generate U~U(0,1) Bucket m = Integer value of (M U), set k = b(m). While (F(k) < U) k = k+1, F(k) = F(k-1) + pk Set N = k The average number of iterations now can be calculated by conditioning on \$U\$ belonging to the intervals of the form [m/M, (m+1)/M). Given the bucket m, N = X(m) + b(m) where X(m) has the corresponding residual distribution. Clearly, E[X(m)] &\lt; E[N] and equality only hapens if all the elements belong to the last bucket, for only then will b(m) = 0 for all m. Buckets are equally probable and the number of iterations required given bucket m is E[X(m)] + 1. Therefore a reduction in computational time can be achieved.

Accelerating Techniques: Memory vs Speed If the random variable N has a finite number of values, it may be more advantageous to calculte the distribution F(k) and keep it in memory, rather than evaluating F(k) = F(k-1) + pk every time we want to generate a random variable.

 Accelerating Techniques: Reordering The order in which the comparisons are made in the algorithm above is always the same, starting with k=0 and going upwards. Suppose that a particular value, say k=5 has a much larger probability than other values. If the inverse funtion method above is used, then most of the time at least five comparisons or iterations of the algorithms have to be made. Instead, we may reorder the values by decreasing probability. A correspondence between two random variables is established, setting a unique function X(N) that assigns to the value N its order, so that X(N) = 0 for the value of N with highest probability, X(N)=k for the (k+1)-st most probable value. The random variable X is then generate by the Inverse Method above, and N is set to the unique corresponding value. The average number of iterations is now E[X]+1 < E[N] +1.
 Methods Distributions