8. Consider the technique of simulating a gamma (, ) random variable by using the rejection method...

8. Consider the technique of simulating a gamma (ç, ë) random variable by using the rejection method with g being an exponential density with rate

ë/ç.

(a) Show that the average number of iterations of the algorithm needed to generate a gamma is nnex~n/(n -1)1.

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

(a) is approximately equal to e[(n - 1)/(2ð)]1 / 2.

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

Step 1: Generate Yx and Y2, independent exponentials with rate 1.

Step 2: If Yx < (n - \)[Y2 - log(Y2) - 1], return to Step 1.

Step 3: Set X = çÕ2/ë.

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