Expectation of Shifted Geometric Distribution/Proof 1

Proof
From the definition of expectation:


 * $\displaystyle E \left({X}\right) = \sum_{x \mathop \in \Omega_X} x \Pr \left({X = x}\right)$

By definition of shifted geometric distribution:
 * $\displaystyle E \left({X}\right) = \sum_{k \mathop \in \Omega_X} k p \left({1 - p}\right)^{k-1}$

Let $q = 1 - p$: