2. Suppose Z1, Z2, . . . , Zk are i.i.d. N(0, 1) random variables and let X = X k i=1 Z 2 i .

a. Show that the moment generating function MZ(2 1) (t) of Z (2 1) is given by (1 2t)^ ( 1 /2) , t is not equal to 1/ 2 .

b. Hence, show using the method of moment generating functions, that the moment generating function of X is MX(t) = 1 / (1 2t)^ k/2 , t is not equal to 1/ 2 . [3 marks]

c. Name the distribution of X. [1 mark]

d. Hence, or otherwise, show that E[X] = k and Var(X) = 2k. [4 marks] [Total: 12]