# Definition:Geometric Distribution/Formulation 1

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## Definition

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

$X$ 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 $0 < p < 1$.

It is written:

- $X \sim \Geometric p$

## Also presented as

The geometric distribution can also be found presented as:

- $\map \Pr {X = k} = p^k q$

where $q = 1 - p$.

## Also see

- Bernoulli Process as Geometric Distribution: the model of the number of successes achieved in a series of Bernoulli trials before the first failure is encountered.

- 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**