Moment generating function of a cumulative discrete distribution

Question

I have the following cumulative distribution function:

    F(x) = 0, if x < 0
        2/8, if 0 <= x < 2
        3/8, if 2 <= x < 4
        1, if >= 4

I have been asked to find the moment generating of function and I am not sure how to do that. Would really appreciate any help!

Note '<=' refers to less than equal to

Answer 1

$F$ is the distribution function of the random variable such that $P(X = 0) = 1/4$, $P(X=2) = 1/8$, $P(X = 4) = 5/8$. Hence, for every $s \in \mathbb{R}$, $$ E[\exp(sX)] = \frac{2+\exp(s) + 5\exp(2s)}{8}. $$