Suppose X is a random variable with MGF Moment Generating Fu

Suppose X is a random variable with MGF (Moment Generating Function) Mx(t)=(1/8)e^t+(1/4)e^2t+(5/8)e^5t.

What is the distribution of X and what is P[X=2]?

Solution

Since MGF = E(e^tX) = sum[ P(X=k) * e^(tk) ], we compare terms to get the PMF:

P(X=1) = 1/8

P(X = 2) = 1/4

P(X = 5) = 5/8

and P(X= x) = 0 for other values.

Thus , P(X=2) = 1/4 = 0.25