Expectation of Geometric Distribution/Proof 2

Theorem
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$.

Then the expectation of $X$ is given by:
 * $E \left({X}\right) = \dfrac p {1-p}$

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


 * $\Pi_X \left({s}\right) = \dfrac q {1 - ps}$

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.