Expectation of Geometric Distribution/Formulation 2/Proof 2

Proof
By Moment Generating Function of Geometric Distribution, the moment generating function 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.

From Moment in terms of Moment Generating Function:


 * $\expect X = \map { {M_X}'} 0$

From Moment Generating Function of Geometric Distribution: First Moment:


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

Hence setting $t = 0$: