Definition:Geometric Distribution

Definition
Let $X$ be a discrete random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

$X$ has the geometric distribution with parameter $p$ iff: where $0 < p < 1$.
 * $X \left({\Omega}\right) = \left\{{0, 1, 2, \ldots}\right\} = \N$
 * $\Pr \left({X = k}\right) = \left({1 - p}\right) p^k$

It is frequently seen as:
 * $\Pr \left({X = k}\right) = p^k q$

where $q = 1 - p$.

It is written:
 * $X \sim \operatorname{G}_0 \left({p}\right)$

Shifted Geometric Distribution
There is a different form of the geometric distribution, as follows:

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: $E \left({X}\right) = \dfrac p {1-p}$


 * Expectation of Shifted Geometric Distribution: $E \left({X}\right) = \dfrac 1 p$

Also, beware confusion: some treatments of this subject define the geometric distribution as the number of failures before the first success, that is: which makes this distribution hardly any different from (and therefore, hardly any more useful than) the shifted geometric distribution.
 * $X \left({\Omega}\right) = \left\{{0, 1, 2, \ldots}\right\} = \N^*$
 * $\Pr \left({X = k}\right) = p \left({1 - p}\right)^k$

Also see

 * Geometric Distribution Gives Rise to Probability Mass Function


 * Expectation of Geometric Distribution


 * 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.