# Moment Generating Function of Geometric Distribution/Formulation 2/Examples

## Examples of Use of Moment Generating Function of Geometric Distribution

Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.

$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
$\map \Pr {X = k} = p \paren {1 - p}^k$

### First Moment

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

$\map { {M_X}'} t = \dfrac {p \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^2 }$

### Second Moment

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

$\map { {M_X}''} t = p \paren {1 - p} e^t \paren {\dfrac {1 + \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^3 } }$

### Third Moment

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

$\map { {M_X}'''} t = p \paren {1 - p} e^t \paren {\dfrac {1 + 4 \paren {1 - p} e^t + \paren {1 - p}^2 e^{2t} } {\paren {1 - \paren {1 - p} e^t}^4 } }$

### Fourth Moment

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

$\map { {M_X}^{\paren 4} } t = p \paren {1 - p } e^t \paren {\dfrac {1 + 11 \paren {1 - p} e^t + 11 \paren {1 - p}^2 e^{2t} + \paren {1 - p}^3 e^{3t} } {\paren {1 - \paren {1 - p} e^t}^5 } }$