Expectation of Geometric Distribution/Proof 2

Proof
From the Probability Generating Function of Geometric Distribution:


 * $\map {\Pi_X} s = \dfrac q {1 - p s}$

where $q = 1 - p$.

From Expectation of Discrete Random Variable from PGF:


 * $\expect X = \map {\Pi'_X} 1$

We have:

Plugging in $s = 1$:

Hence the result.