Moment Generating Function of Geometric Distribution/Formulation 1/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.

$\map X \Omega = \set {0, 1, 2, \ldots} = \N$

$\map \Pr {X = k} = \paren {1 - p} p^k$

First Moment

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

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

Second Moment

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

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

Third Moment

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

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

Fourth Moment

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

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