Exercise: Sum of Two Independent Poissons via CFs

Prerequisites: Characteristic Functions, Poisson Distribution

Problem

A credit-risk model tracks defaults arriving at two independent intensities: investment-grade defaults $N_1 \sim \text{Poisson}(\lambda_1)$ and high-yield defaults $N_2 \sim \text{Poisson}(\lambda_2)$. Let $N = N_1 + N_2$ be the total.

1. Using CFs, show that $N \sim \text{Poisson}(\lambda_1 + \lambda_2)$.

2. Contrast with the case of non-independent $N_1, N_2$. Suppose instead $N_1 = N_2 = N^*$ where $N^* \sim \text{Poisson}(\lambda)$. Compute $\varphi_{N_1 + N_2}(t)$ and show the result is not a Poisson CF. What distribution is $N_1 + N_2$ in this case?
3. Extension. In the independent case, given $N_1 + N_2 = n$, what is the conditional distribution of $N_1$? Derive it using the CF of conditional distributions (or directly; the CF route is surprisingly clean).

Hint

Part 1 is almost immediate from the CF convolution rule. For part 2, $N_1 + N_2 = 2N^*$ — no longer Poisson! For part 3, the conditional distribution given the sum is binomial.