Let X have an exponential distribution with 1. (a) Find the first four cumulants of

Let X have an exponential distribution with ¸ Æ 1.

(a) Find the first four cumulants of the distribution of X.

(b) Use the expressions in Section 8.4 to calculate the third and fourth moments of Zn Æ

p n

³

Xn ¡¹

´

for n Æ 10, n Æ 100, n Æ 1000.

(c) How large does n be in order to for the third and fourth moments to be “close” to those of the normal approximation. (Use your judgment to assess closeness.)