# Category:Moment Generating Function of Gaussian Distribution

This category contains pages concerning Moment Generating Function of Gaussian Distribution:

## Theorem

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

Then the moment generating function $M_X$ of $X$ is given by:

$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

## Proof

From the definition of the Gaussian distribution, $X$ has probability density function:

$\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$

From the definition of a moment generating function:

$\ds \map {M_X} t = \expect { e^{t X} } = \int_{-\infty}^\infty e^{t x} \map {f_X} x \rd x$

So:

 $\ds \map {M_X} t$ $=$ $\ds \frac 1 {\sigma \sqrt{2 \pi} } \int_{-\infty}^\infty \map \exp {t x - \frac {\paren {x - \mu}^2} {2 \sigma^2} } \rd x$ $\ds$ $=$ $\ds \frac {\sqrt 2 \sigma} {\sigma \sqrt {2 \pi} } \int_{-\infty}^\infty \map \exp {\paren {\sqrt 2 \sigma u + \mu} t - u^2} \rd u$ substituting $u = \dfrac {x - \mu} {\sqrt 2 \sigma}$ $\ds$ $=$ $\ds \frac {\exp \mu t} {\sqrt \pi} \int_{-\infty}^\infty \map \exp {-\paren {u^2 - \sqrt 2 \sigma u t} } \rd u$ $\ds$ $=$ $\ds \frac {\exp \mu t} {\sqrt \pi} \int_{-\infty}^\infty \map \exp {-\paren {u - \frac {\sqrt 2} 2 \sigma t}^2 + \frac 1 2 \sigma^2 t^2} \rd u$ $\ds$ $=$ $\ds \frac {\map \exp {\mu t + \frac 1 2 \sigma^2 t^2} } {\sqrt \pi} \int_{-\infty}^\infty \map \exp {-v^2} \rd v$ substituting $v = u - \dfrac {\sqrt 2} 2 \sigma t$ $\ds$ $=$ $\ds \frac {\sqrt \pi \map \exp {\mu t + \frac 1 2 \sigma^2 t^2} } {\sqrt \pi}$ Gaussian Integral $\ds$ $=$ $\ds \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

$\blacksquare$

## Examples

### First Moment

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

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

### Second Moment

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

$\map { {M_X}} t = \paren {\sigma^2 + \paren {\mu + \sigma^2 t}^2 } \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

### Third Moment

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

$\map { {M_X}} t = \paren {3 \sigma^2 \paren {\mu + \sigma^2 t} + \paren {\mu + \sigma^2 t}^3} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

### Fourth Moment

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

$\map { {M_X}^{\paren 4} } t = \paren {3 \sigma^4 + 6 \sigma^2 \paren {\mu + \sigma^2 t}^2 + \paren {\mu + \sigma^2 t}^4} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

## Subcategories

This category has only the following subcategory.

## Pages in category "Moment Generating Function of Gaussian Distribution"

The following 6 pages are in this category, out of 6 total.