Variance of Geometric Distribution

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Theorem

Let $p \in \R$ be a real number such that $0 < p < 1$.

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


Formulation 1

$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
$\map \Pr {X = k} = \paren {1 - p} p^k$


Then the variance of $X$ is given by:

$\var X = \dfrac p {\paren {1-p}^2}$


Formulation 2

$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
$\map \Pr {X = k} = p \paren {1 - p}^k$


Then the variance of $X$ is given by:

$\var X = \dfrac {1 - p} {p^2}$