Moment generating function - Can it be a polynomial? Let X be a random variable such that P(X = 0) < 1. 
Is it possible for a moment generating function of X to be a polynomial $a_n t^n + a_{n-1} t^{n-1} +\cdots+  a_1 t +  a_0$? Why

If the mgf $M_X(t)$ is a polynomial, the $2k$-th derivative of $M_X(t)$ is identically $0$ for some positive integer $k$. It follows that $E(X^{2k})=0$, which is only possible if $X$ is $0$ with probability $1$.
