# Definition:Geometric Distribution

## 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 frequently seen as:

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

where $q = 1 - p$.

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 frequently seen as:

- $\map \Pr {X = k} = q^{k-1} p$

where $q = 1 - p$.

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, **beware confusion**: some treatments of this subject define the **geometric distribution** as the number of failures before the first success, that is:

- $\map X \Omega = \set {0, 1, 2, \ldots} = \N$
- $\map \Pr {X = k} = p \paren {1 - p}^k$

which makes this distribution hardly any different from (and therefore, hardly any more useful than) the **shifted geometric distribution**.

## Also see

- Bernoulli Process as Geometric Distribution, where it is shown that this models 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`