Expectation of Geometric Distribution/Formulation 2/Proof 1

Proof
From the definition of expectation:


 * $\ds \expect X = \sum_{x \mathop \in \Omega_X} x \map \Pr {X = x}$

By definition of geometric distribution:


 * $\ds \expect X = \sum_{k \mathop \in \Omega_X} k p \paren {1 - p}^k$

Let $q = 1 - p$: