Poisson Limit of the Binomial via Characteristic Functions

Let $X_n \sim \text{Binomial}(n, \lambda/n)$ for fixed $\lambda > 0$ and $n > \lambda$.

(a) Write down the characteristic function $\varphi_{X_n}(t) = E[e^{itX_n}]$ in closed form.

(b) Show that $\lim_{n \to \infty} \varphi_{X_n}(t) = e^{\lambda(e^{it} - 1)}$ for every $t \in \mathbb{R}$.

(c) Identify the limiting characteristic function and state the convergence-in-distribution conclusion.

(d) Justify why pointwise convergence of characteristic functions implies convergence in distribution (cite the relevant theorem).

(e) For $\lambda = 5$, $n = 100$: compute $P(X_n = 3)$ using both the exact Binomial PMF and the Poisson approximation, and find the relative error.