Definition:Geometric Distribution

Definition

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

There are two forms of the geometric distribution, as follows.

Geometric Distribution

$X$ has the geometric distribution with parameter $p$ (where $0 < p < 1$) if:

• $\Omega \left({X}\right) = \left\{{0, 1, 2, \ldots}\right\} = \N^*$

• $\Pr \left({X = k}\right) = \left({1 - p}\right) p^k$

It is frequently seen as:

$\Pr \left({X = k}\right) = p^k q$

where $q = 1 - p$.

It is written:

$X \sim \operatorname{G}_0 \left({p}\right)$

It can be seen in Bernoulli Process as a Geometric Distribution that this models the number of successes achieved in a series of Bernoulli trials before the first failure is encountered.

Shifted Geometric Distribution

$X$ has the shifted geometric distribution with parameter $p$ (where $0 < p < 1$) if:

• $\operatorname{Im} \left({X}\right) = \left\{{1, 2, \ldots}\right\} = \N^*$

• $\Pr \left({X = k}\right) = p \left({1 - p}\right)^{k-1}$

It is frequently seen as:

$\Pr \left({X = k}\right) = q^{k-1} p$

where $q = 1 - p$.

It is written:

$X \sim \operatorname{G}_1 \left({p}\right)$

It can be seen in Bernoulli Process as a Shifted Geometric Distribution that this models the number of Bernoulli trials performed before the first success is achieved.

Its Probability Mass Function

Note that the Geometric Distribution Gives Rise to Probability Mass Function satisfying $\Pr \left({\Omega}\right) = 1$.

Note

The distinction may appear subtle, but the two distributions do have subtly 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:

• $\Omega \left({X}\right) = \left\{{0, 1, 2, \ldots}\right\} = \N^*$
• $\Pr \left({X = k}\right) = p \left({1 - p}\right)^k$

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

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 2.2$ where a shifted geometric distribution is called a geometric distribution.