# 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

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

- 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**