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:


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


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

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

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

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

Note
The distinction may appear relatively trivial, but the two distributions do have subtly different behaviour.

For example (and perhaps most significantly), their expectations are different:


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


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