Moment Generating Function of Geometric Distribution/Formulation 2/Examples/Third 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 third moment generating function of $X$ is given by:


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

Proof
We have: