Expectation of Shifted Geometric Distribution/Proof 2

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


 * $\displaystyle \Pi_X \left({s}\right) = \frac {ps} {1 - qs}$

where $q = 1 - p$.

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


 * $E \left({X}\right) = \Pi'_X \left({1}\right)$

We have:

Plugging in $s = 1$:

Hence the result.