> [!NOTE] Theorem > Let $p\in(0,1].$ Let $X\sim \text{Geom}(p)$ where $\text{Geom}(p)$ denotes a [[Geometric Distribution|geometric distribution]] with parameter $p.$ Then the [[Moment generating function of real-valued random variable|moment generating function]] of $X$ is given by $M_{X}(t)=\begin{cases} \dfrac{p}{e^{-t}+p-1}, & t < \ln \frac{1}{1-p} \\ \infty, & t \geq \ln \frac{1}{1-p}. \end{cases}$ **Proof**: ....