Expectation of Shifted Geometric Distribution/Proof 2

Proof
From the Probability Generating Function of Shifted Geometric Distribution, we have:


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

where $q = 1 - p$.

From Expectation of Discrete Random Variable from PGF, we have:


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

We have:

Plugging in $s = 1$:

Hence the result.