Moment Generating Functions

Definition

The moment generating function (MGF), or Laplace Transform, of $X$ is defined by ${\psi }_X(t)=\mathbb{E}\left(e^{tX}\right)=\int e^{tx}dF(x)$ where $t$ varies over the real numbers.

When the MGF is well defined, it can be shown that we can interchange the operations of differentiation and expectation. This leads to ${\psi }^{\prime }(0)={\left[\frac{d}{dt}\mathbb{E}{e}^{tX}\right]}_{t=0}=\mathbb{E}{\left[\frac{d}{dt}{e}^{tX}\right]}_{t=0}=\mathbb{E}{\left[X{e}^{tx}\right]}_{t=0}=\mathbb{E}(X).$

By taking $k$ derivatives, we conclude that ${\psi }^{(k)}(0)=\mathbb{E}(X^k)$. This gives us a method for computing the moments of a distribution.

Lemma (Properties of the MFG)

1. If $Y=aX+b$, then ${\psi }_Y(t)=e^{bt}{\psi }_X(at)$
2. If $X_1,\dots ,X_n$ are independent and $Y={\sum }_i{X}_i$, Then ${\psi }_Y(t)={\prod }_i{\psi }_i(t)$, where ${\psi }_i$ is the MGF of $X_i$.

Theorem

Let $X$ and $Y$ be random variables. If ${\psi }_X(t)={\psi }_Y(t)$ for all $t$ is an open interval around 0, then $X\stackrel{d}{=}Y$.