1. (a) Find the moment-generating function for a gamma probability distribution with parameter α > 0 and β > 0. [Hint: In the integral representation of E(etX), change the variable t to u = (1 - βt)x/β, with (1 - βt) > 0.]

(b) Using the mgf of a gamma probability distribution, ?nd E(X) and Var(X).

2. Let X be an exponential random variable. Show that, for numbers a > 0 and b > 0,

P(X > a + b |X > a) = P(X > b).

(This property of the exponential distribution is called the memoryless property of the distribution.)