Definition:Geometric Distribution/Shifted

Definition
Let $X$ be a discrete random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. $X$ has the shifted geometric distribution with parameter $p$ iff: where $0 < p < 1$.
 * $X \left({\Omega}\right) = \left\{{1, 2, \ldots}\right\} = \N_{>0}$
 * $\Pr \left({X = k}\right) = p \left({1 - p}\right)^{k-1}$

It is frequently seen as:
 * $\Pr \left({X = k}\right) = q^{k-1} p$

where $q = 1 - p$.

It is written:
 * $X \sim \operatorname{G}_1 \left({p}\right)$

Also known as
Some sources, for example, refer to this as the geometric distribution, failing adequately to distinguish between this and what is defined on as the geometric distribution.

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: 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

 * Definition:Geometric Distribution


 * Expectation of Shifted Geometric Distribution


 * Bernoulli Process as Shifted Geometric Distribution where it is shown that this models the number of Bernoulli trials performed before the first success is achieved.