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