Exercise: MGF Computation — Exponential and Gamma

Prerequisites: Moment Generating Functions

Problem

Compute MGFs directly from the definition $M_X(s) = \mathbb{E}[e^{sX}]$ for two important distributions.

1. Let $X \sim \text{Exp}(\lambda)$ with density $f_X(x) = \lambda e^{-\lambda x}$, $x \ge 0$. Compute $M_X(s)$ and state its domain.

2. From $M_X(s)$, compute $\mathbb{E}[X]$ and $\text{Var}(X)$ by differentiation. Verify your answers against the known formulas $\mathbb{E}[X] = 1/\lambda$, $\text{Var}(X) = 1/\lambda^2$.

3. Let $Y_1, \ldots, Y_n$ be i.i.d. $\text{Exp}(\lambda)$. Use the product-of-MGFs rule to show that $Y := Y_1 + \cdots + Y_n$ has MGF $M_Y(s) = \left(\frac{\lambda}{\lambda - s}\right)^n, \qquad s < \lambda.$

4. Identify the distribution of $Y$ by recognising this as the MGF of a named family. (Hint: it's a gamma distribution.)

Hint

The integral $\int_0^\infty e^{(s - \lambda)x}\,dx$ converges iff $s < \lambda$, which gives the domain.