The mgf of a certain random variable $X$ is $M_X(t)=e^{\frac12(e^t -1)}$. Find $\mathbb P(X>0)$.
-I know that $X\sim\mathrm{Poi}(1/2)$.
-I also know I solve this equation as $1-\mathbb P(X=0)$.
-I plugged $0$ to $t$ in the equation but my result was $0$, which I don't believe is true.
Please Help!
Actually $M_X(0)=1$, as is the case for all moment generating functions, but we cannot easily recover the probability distribution of $X$ from its moment generating function. Instead, recognizing that $X\sim\mathrm{Pois}(1/2)$, we find that $$ \mathbb P(X>0) = 1-\mathbb P(X=0) = 1-e^{-1/2}. $$