# Definition:Geometric Distribution/Formulation 2

## Definition

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

$X$ has the geometric distribution with parameter $p$ if and only if:

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

where $0 < p < 1$.

It is written:

$X \sim \Geometric p$

## Also presented as

The geometric distribution can also be found presented as:

$\map \Pr {X = k} = p q^k$

where $q = 1 - p$.

## Also see

• Results about the geometric distribution can be found here.

## Technical Note

The $\LaTeX$ code for $\Geometric {p}$ is \Geometric {p} .

When the argument is a single character, it is usual to omit the braces:

\Geometric p