Question

Let X be a nonnegative integer valued random variable with pmf
P(k)={ 1/2, k=0
(1/3)^k k=1,2,3,... }
a) Find the moment generating function of X, for all possible real values of its argument t. Express your answer in the most compact form possible.
(ie, Indicate for what values of t, the infinite series or improper integral is convergent)
b) Determine E(x) and Var(x).