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

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

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

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

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

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

## Proof

We have:

 $\ds \map { {M_X}^{\paren 4} } t$ $=$ $\ds \frac \d {\d t} \map { {M_X}'''} t$ Definition of Moment Generating Function $\ds$ $=$ $\ds \frac \d {\d t} p \paren {1 - p} \paren {\dfrac {e^t + 4 \paren {1 - p} e^{2t} + \paren {1 - p}^2 e^{3t} } {\paren {1 - \paren {1 - p} e^t}^4 } }$ Moment Generating Function of Geometric Distribution: Third Moment $\ds$ $=$ $\ds p \paren {1 - p} \frac \d {\d t} \paren {e^t + 4 \paren {1 - p} e^{2t} + \paren {1 - p}^2 e^{3t} } \paren {1 - \paren {1 - p} e^t}^{-4}$ factoring out the $p \paren {1 - p}$ $\ds$ $=$ $\ds p \paren {1 - p } \paren {\paren {e^t + 8 \paren {1 - p} e^{2t} + 3 \paren {1 - p}^2 e^{3t} } \paren {1 - \paren {1 - p} e^t}^{-4} + \paren {e^t + 4 \paren {1 - p} e^{2t} + \paren {1 - p}^2 e^{3t} } \paren {-4 \paren {1 - \paren {1 - p} e^t}^{-5} } \paren {-\paren {1 - p} e^t } }$ Product Rule, Chain Rule for Derivatives, Derivative of Power, Derivative of Exponential Function $\ds$ $=$ $\ds p \paren {1 - p } \paren {\dfrac {e^t + 8 \paren {1 - p} e^{2t} + 3 \paren {1 - p}^2 e^{3t} } {\paren {1 - \paren {1 - p} e^t}^4 } + \dfrac {4 \paren {1 - p} e^{2t} + 16 \paren {1 - p}^2 e^{3t} + 4 \paren {1 - p}^3 e^{4t} } {\paren {1 - \paren {1 - p} e^t}^5 } }$ gathering terms $\ds$ $=$ $\ds p \paren {1 - p } \paren {\dfrac {e^t + 8 \paren {1 - p} e^{2t} + 3 \paren {1 - p}^2 e^{3t} } {\paren {1 - \paren {1 - p} e^t}^4 } \dfrac {\paren {1 - \paren {1 - p} e^t} } {\paren {1 - \paren {1 - p} e^t} } + \dfrac {4 \paren {1 - p} e^{2t} + 16 \paren {1 - p}^2 e^{3t} + 4 \paren {1 - p}^3 e^{4t} } {\paren {1 - \paren {1 - p} e^t}^5 } }$ multiplying by $1$ $\ds$ $=$ $\ds p \paren {1 - p } \dfrac {e^t + 8 \paren {1 - p} e^{2t} + 3 \paren {1 - p}^2 e^{3t} - \paren {1 - p} e^{2t} - 8 \paren {1 - p}^2 e^{3t} - 3 \paren {1 - p}^3 e^{4t} + 4 \paren {1 - p} e^{2t} + 16 \paren {1 - p}^2 e^{3t} + 4 \paren {1 - p}^3 e^{4t} } {\paren {1 - \paren {1 - p} e^t}^5 }$ simplifying $\ds$ $=$ $\ds p \paren {1 - p } e^t \paren {\dfrac {1 + 11 \paren {1 - p} e^t + 11 \paren {1 - p}^2 e^{2t} + \paren {1 - p}^3 e^{3t} } {\paren {1 - \paren {1 - p} e^t}^5 } }$ simplifying

$\blacksquare$