Definition:Geometric Distribution/Formulation 1

Definition
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. $X$ has the geometric distribution with parameter $p$ :
 * $\map X \Omega = \set {0, 1, 2, \ldots} = \N$
 * $\map \Pr {X = k} = \paren {1 - p} p^k$

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.


 * Definition:Geometric Distribution/Formulation 2