Expectation of Geometric Distribution/Formulation 1

Theorem
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$ for some $0 < p < 1$.
 * $\map X \Omega = \set {0, 1, 2, \ldots} = \N$
 * $\map \Pr {X = k} = \paren {1 - p} p^k$

Then the expectation of $X$ is given by:
 * $\expect X = \dfrac p {1 - p}$