83. Simulation studies are important in investigating various characteristics of a system or process. They are generally employed when the mathematical analysis necessary to answer important questions is too complicated to yield closed form solutions. For example, in a system where the time between successive customer arrivals has a particular pdf and the service time of any particular customer has another particular pdf, simulation can provide information about the probability that the system is empty when a customer arrives, the expected number of customers in the system, and the expected waiting time in queue. Such studies depend on being able to generate observations from a speci ed probability distribution.

The rejection method gives a way of generating an observation from a pdf f (#) when we have a way of generating an observation from g(#) and the ratio f (x)/g(x) is bounded, that is, c for some nite c.

The steps are as follows:

1. Use a software package s random number generator to obtain a value u from a uniform distribution on the interval from 0 to 1.

2. Generate a value y from the distribution with pdf g(y).

3. If u f (y)/cg(y), set xy ( accept x); otherwise return to step 1. That is, the procedure is repeated until at some stage u f (y)/cg(y).

a. Argue that c  1. Hint: If c  1, then f (y)  g(y)

for all y; why is this bad?

b. Show that this procedure does result in an observation from the pdf f (); that is, P(accepted value x)  F(x). Hint: This probability is P({U f (Y)/cg(Y)}  {Y x}); to calculate, rst integrate with respect to u for xed y and then integrate with respect to y.

c. Show that the probability of accepting at any particular stage is 1/c. What does this imply about the expected number of stages necessary to obtain an acceptable value? What kind of value of c is desirable?

d. Let f (x)  20x(1  x)3 for 0  x  1, a particular beta distribution. Show that taking g(y) to be a uniform pdf on (0, 1) works. What is the best value of c in this situation?