8. Consider the technique of simulating a gamma random variable by using the rejection method with g being an exponential density with rateI mage.

(a) Show that the average number of iterations of the algorithm needed to generate a gamma is Image.

(b) Use Stirling's approximation to show that for large n the answer to part

(a) is approximately equal to Image.

(c) Show that the procedure is equivalent to the following:

Step 1: Generate and , independent exponentials with rate 1.

Step 2: IfI mage, return to step 1.

Step 3: SetI mage.

(d) Explain how to obtain an independent exponential along with a gamma from the preceding algorithm.