Exercise: Sum of Independent Poissons via MGFs

Prerequisites: Moment Generating Functions

Problem

Let $X \sim \text{Poisson}(\lambda_1)$ and $Y \sim \text{Poisson}(\lambda_2)$ be independent.

1. Compute $M_X(s)$ directly from the definition (use the Poisson pmf $\mathbb{P}(X = k) = e^{-\lambda_1}\lambda_1^k / k!$).

2. Use the product-of-MGFs rule to show $M_{X + Y}(s) = \exp((\lambda_1 + \lambda_2)(e^s - 1))$.

3. Identify the distribution of $X + Y$ by recognising the MGF as belonging to a named family. State the parameters.

4. Market-microstructure application. Suppose buy-side order arrivals follow a Poisson process with rate $\lambda_1 = 20$ per minute and sell-side arrivals follow an independent Poisson process with rate $\lambda_2 = 15$ per minute. What is the distribution of the total number of orders per minute, and what is its mean and variance? Use the MGF result.

Hint

For part 1: $M_X(s) = \sum_{k=0}^\infty e^{sk}\,e^{-\lambda_1}\lambda_1^k/k! = e^{-\lambda_1}\sum_{k=0}^\infty (\lambda_1 e^s)^k/k! = e^{-\lambda_1}\,e^{\lambda_1 e^s}$. Simplify.