Moment Generating Function of Geometric Distribution

Theorem

Let $X$ be a discrete random variable with a geometric distribution with parameter $p$ for some $0 < p < 1$.

Formulation 1

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

Then the moment generating function $M_X$ of $X$ is given by:

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

for $t < -\map \ln p$, and is undefined otherwise.

Formulation 2

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

Then the moment generating function $M_X$ of $X$ is given by:

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

for $t < -\map \ln {1 - p}$, and is undefined otherwise.