Definition:Geometric Distribution

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Definition

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

Then $X$ obeys a geometric distribution if and only if $\map Pr {X = k}$ decreases in geometric progression as $k$ increases.


There are vrious formulations of the geometric distribution:


Formulation 1

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

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

where $0 < p < 1$.


Formulation 2

$X$ 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:

$X \sim \Geometric p$


Shifted Geometric Distribution

There is a different form of the geometric distribution, as follows:

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

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

where $0 < p < 1$.


It is written:

$X \sim \ShiftedGeometric p$


Note


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The distinction between this and the shifted geometric distribution may appear subtle, but the two distributions do have different behaviour.

For example (and perhaps most significantly), their expectations are different:


Expectation of Geometric Distribution: $\expect X = \dfrac p {1 - p}$
Expectation of Shifted Geometric Distribution: $\expect X = \dfrac 1 p$


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


Sources


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