Definition:Geometric Distribution
Definition
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
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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- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: geometric distribution
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: geometric distribution
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: geometric distribution