Let the random variables in the sequence \(\left\{X_{n}ight\}\) be iid with a gamma density having parameters \(\alpha=2\) and \(\beta=3\).

(a) What is the probability density for \(\bar{X}_{n}\) ?

(b) What is the asymptotic probability density for \(\bar{X}_{n}\) ?

(c) Plot the actual versus asymptotic probability density for \(\bar{X}_{n}\) when \(n=10\).

(d) Repeat

(c) for \(n=40\). Interpret the graphs in

(c) and

(d) in terms of asymptotic theory.