Moment Generating Function of Geometric Distribution

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

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


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

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

Proof
From the definition of the geometric distribution, $X$ has probability mass function:


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

From the definition of a moment generating function:


 * $\ds \map {M_X} t = \expect {e^{t X} } = \sum_{k \mathop = 0}^\infty \map \Pr {X = k} e^{k t}$

So:

By Sum of Infinite Geometric Sequence, for this sum to be convergent we must have:


 * $\size {p e^t} < 1$

In the case $p = 0$, this demand is satisfied immediately regardless of $t$.

Otherwise, as both $e^t$ and $p$ are positive:


 * $e^t < \dfrac 1 p$

So, by Logarithm of Power:


 * $t < -\map \ln p$

is the range of $t$ for which the expectation is well-defined.

Now applying Sum of Infinite Geometric Sequence, we have for this range of $t$:


 * $\ds \map {M_X} t = \paren {1 - p} \sum_{k \mathop = 0}^\infty \paren {p e^t}^k = \frac {1 - p} {1 - p e^t}$