Probability from the moment generating function

Question

Let $\lambda>0$ be fixed and let $X$ be a random variable with the moment generating function $M_X(t)$ given by

$M_X(t)$=$1$+$\frac{1}{e^\lambda}$($e^t$ $-1$)

How do we find the probability $\mathbb{P}$($X=0$)?

Any help is much appreciated!

Answer 1

For $X$ taking non-negative integer values we have $M_X(t)=\sum P(X=n) e^{nt}$. In this case $M_X(t)=a+be^{t}$ where $a=1-e^{-\lambda}$ and $b=e^{-\lambda}$. Hence, $P(X=0)= a$, $P(X=1)=b$ and $P(X=n)=0$ for all $n \geq 2$. (The fact that $X$ is integer valued follows by uniquness of the distribution for agiven MGF).