Bernoulli Distribution: Moments and MGF

Let $X \sim \text{Bernoulli}(p)$, so $P(X=1)=p$ and $P(X=0)=1-p$ where $0 < p < 1$.

(a) Compute $E[X]$ and $E[X^2]$ directly from the PMF. Hence find $\text{Var}(X)$.

(b) Derive the moment-generating function $M_X(t) = E[e^{tX}]$ and verify that $M_X'(0) = E[X]$ and $M_X''(0) = E[X^2]$.

(c) Show that $\text{Var}(X) = p(1-p) \le 1/4$ for all $p \in (0,1)$, and identify the value of $p$ that maximizes the variance.

(d) If $S = \sum_{i=1}^{n} X_i$ where $X_1, \ldots, X_n$ are iid $\text{Bernoulli}(p)$, use the MGF to show $S \sim \text{Binomial}(n, p)$.