# 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**