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

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

(a) Show that the average number of iterations of the algorithm needed to generate a gamma is nne1−n/(n − 1)!.

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

(a) is approximately equal to e[(n − 1)/(2π)]1/2.

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

Step 1: Generate Y1 and Y2, independent exponentials with rate 1.

Step 2: If Y1 < (n − 1)[Y2 − log(Y2) − 1], return to step 1.

Step 3: Set X = nY2/λ.

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