Exercise: Computing CFs of Standard Distributions by Integration

Prerequisites: Characteristic Functions, Normal Distribution, Exponential Distribution

Problem

1. Derive the CF of $X \sim \mathcal{N}(\mu, \sigma^2)$ by completing the square in the integral $\int e^{itx}\cdot\phi(x;\mu,\sigma^2)\,dx$. (Warning: the "complete the square" trick with complex arguments requires a brief contour argument; stating it at the end is enough.)

2. Derive the CF of $X \sim \text{Exp}(\lambda)$ by direct integration: $\varphi_X(t) = \int_0^\infty e^{itx}\lambda e^{-\lambda x}\,dx$.

3. Derive the CF of $X \sim \text{Poisson}(\lambda)$ by summing the series: $\varphi_X(t) = \sum_{k=0}^\infty e^{itk}\cdot e^{-\lambda}\lambda^k/k!$.

4. Using your answer to (3) and the fact that a sum of independent Poissons is Poisson, state and verify the CF-convolution rule $\varphi_{X_1 + X_2}(t) = \varphi_{X_1}(t)\varphi_{X_2}(t)$ for $X_i \sim \text{Poisson}(\lambda_i)$.

Hint

For (1), write $e^{itx}e^{-(x-\mu)^2/(2\sigma^2)}$ and complete the square in $x$. For (3), recognise $\sum_{k} (\lambda e^{it})^k/k! = \exp(\lambda e^{it})$.