Calculating an expectation given the mgf

Question

Hi all, having a bit of difficulty being able to solve the problem above.

I know from the mgf that it's a geometric distribution (p = 0.25, q = 0.75) and am able to find its subsequent pmf. However, when I try to find the expectation given, the series I am trying to solve is diverging. Not sure if this is supposed to happen or I am on the wrong track.

Thanks!

Answer 1

I'm assuming $M(t)$ is the mgf of $X$, i.e. $M(t) = E(e^{tX})$. If so then $$ E[(1 - 0.75e)e^{X - 1} + 0.75] = \frac{1 - 0.75 e}{e} E[e^X] + 0.75 $$ But $M(1) = \infty$ (the values on which a mgf is finite forms a convex set therefore since it is infinite at $t = - \ln(0.75)$ is can't be finite at $1$).