Definition:Geometric Distribution

From ProofWiki
Jump to navigation Jump to search

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


This page has been identified as a candidate for refactoring.
Extract this and put it into a separate page as a result, then reference it in Also see.
Until this has been finished, please leave {{Refactor}} in the code.

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.

Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.


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


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Geometric_Distribution&oldid=516394"