Moment Generating Function of Geometric Distribution/Formulation 1/Examples/Fourth Moment

Examples of Use of Moment Generating Function of Geometric Distribution/Formulation 1
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
 * $\map X \Omega = \set {0, 1, 2, \ldots} = \N$
 * $\map \Pr {X = k} = \paren {1 - p} p^k$

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


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

Proof
We have: