Moment Generating Function of Gaussian Distribution/Examples/First Moment

Examples of Use of Moment Generating Function of Gaussian Distribution

Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution.

The first moment generating function of $X$ is given by:

$\map { {M_X}'} t = \paren {\mu + \sigma^2 t} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

Proof

We have:

\(\ds \map { {M_X}'} t\)

\(=\)

\(\ds \frac \d {\d t} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}\)

\(\ds \)

\(=\)

\(\ds \map {\frac \d {\d t} } {\mu t + \frac 1 2 \sigma^2 t^2} \frac \d {\map \d {\mu t + \dfrac 1 2 \sigma^2 t^2} } \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \paren {\mu + \sigma^2 t} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}\)

Derivative of Power, Derivative of Exponential Function

$\blacksquare$