Definition:Geometric Distribution

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

Formulation 2
It is written:
 * $X \sim \Geometric p$

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: $\expect X = \dfrac p {1 - p}$


 * Expectation of Shifted Geometric Distribution: $\expect X = \dfrac 1 p$

Also see

 * Geometric Distribution Gives Rise to Probability Mass Function


 * Expectation of Geometric Distribution