Compound Poisson Distribution: MGF and Moments

Let $N \sim \text{Poisson}(\lambda)$ and let $X_1, X_2, \ldots$ be iid discrete random variables (independent of $N$) with PMF $P(X_i = j) = p_j$ for $j = 1, 2, \ldots$ and MGF $M_X(t) = E[e^{tX_1}]$. Define the compound Poisson sum $S = \sum_{i=1}^{N} X_i$ (with $S = 0$ when $N = 0$).

(a) Derive the MGF of $S$. Show that $M_S(t) = \exp\!\big(\lambda(M_X(t) - 1)\big)$.

(b) Use the MGF to derive $E[S]$ and $\text{Var}(S)$. Express your answers in terms of $\lambda$, $E[X_1]$, and $\text{Var}(X_1)$.

(c) Alternatively, derive $E[S]$ and $\text{Var}(S)$ using the tower property (law of total expectation) and the Eve's law (law of total variance), conditioning on $N$.

(d) Application: An insurance company receives claims at a Poisson rate of $\lambda = 10$ per day. Each claim size is $\$1000$ with probability $0.6$ or $\$5000$ with probability $0.4$, independently. Find $E[S]$ and $\text{Var}(S)$ for the total daily claims $S$, and compute the standard deviation.