Expectation of Geometric Distribution

Theorem

Let $X$ be a discrete random variable with the geometric distribution with parameter $p$ for some $0 < p < 1$.

Formulation 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:

$\map E X = \dfrac p {1-p}$

Formulation 2

$\map X \Omega = \set {0, 1, 2, \ldots} = \N$

$\map \Pr {X = k} = p \paren {1 - p}^k$

Then the expectation of $X$ is given by:

$\map E X = \dfrac {1-p} p$