Example iid variables $X_i$ where $S=\sum_{j=1}^NX_j$ but $M_S(t) \neq P_N(M_X(t))$

Question

Let let $N$ a discrete random variable with support contained in $\mathbb{N}$. If for a fixed value of $N$ we have that $X_1, \ldots, X_N$ are independent and identically distributed random variables with common distribution $X$ which is independent of $N$, then for moment generating function $M_S$ of the variable $$S=\sum_{j=1}^NX_j$$ we have that $$M_S(t)=P_N(M_X(t)),$$ where $M_X$ is the moment generating function of $X$ and $P_N$ is the probability generating function of $N$. I need an example where $X_1, \ldots, X_N$ are independent and identically distributed random variables but with common distribution $X$ which depends of $N$ such that $M_S(t)\neq P_N(M_X(t)).$ Can you help me, please?

Answer 1

Let $\{X_i\}$ be i.i.d taking values $0$ and $1$ with probability $\frac 1 2$ each. Let $N=X_1+1$. Then $M_X(t)=\frac {1+e^{t}} 2$ and $P_N(s)=\frac {s+s^{2}} 2$. I will let you compute $Ee^{t \sum\limits_{k=1}^{N}X_i}$ and show that this quantity is not equal to $P_N(M_X(t))$.