For a positive integer $N, p \in[0,1]$, and $k \in\{0,1, \ldots, N\}$, let

$$p_{k}(N, p)=\left(\begin{array}{c} N \\ k \end{array}\right) p^{k}(1-p)^{N-k}$$

(a) For fixed $N$ and $p$, show that $p_{k}(N, p)$ is a probability mass function on $\{0,1, \ldots, N\}$ and that the corresponding probability distribution has mean $N p$ and variance $N p(1-p)$.

(b) Let $\lambda>0$. Show that, for any $k \in\{0,1,2, \ldots\}$,

$$\lim _{N \rightarrow \infty} p_{k}(N, \lambda / N)=\frac{e^{-\lambda} \lambda^{k}}{k !}$$

Show that the right-hand side of $(*)$ is a probability mass function on $\{0,1,2, \ldots\}$.

(c) Let $p \in(0,1)$ and let $a, b \in \mathbb{R}$ with $a<b$. For all $N$, find integers $k_{a}(N)$ and $k_{b}(N)$ such that

$$\sum_{k=k_{a}(N)}^{k_{b}(N)} p_{k}(N, p) \rightarrow \frac{1}{\sqrt{2 \pi}} \int_{a}^{b} e^{-\frac{1}{2} x^{2}} d x \quad \text { as } N \rightarrow \infty$$

[You may use the Central Limit Theorem.]