# What is the formula for the variance of a geometric distribution?

Jan 22, 2016

${\sigma}^{2} = \frac{1 - p}{p} ^ 2$

#### Explanation:

A geometric probability distribution describes one of the two ‘discrete’ probability situations. In statistics and probability subjects this situation is better known as binomial probability. Thus a geometric distribution is related to binomial probability.
consider a case of binomial trial. As we know already, the trial has only two outcomes, a success or a failure. Let ‘p’ the probability of success and ‘q’ be the probability of failure. Obviously, in case of binomial probability, p = 1 – q.

now suppose the trial is conducted ‘X’ number of times and the first success occurs on kth time. If the probability of such occurrence can be expressed as some geometric function (gdf) of ‘p’ then the probability distribution is called geometric probability distribution.
In other words, P(X = k) = gdf (p) with:
Mean $\mu = \frac{1 - p}{p}$; and standard deviation sigma = sqrt((1-p)/p^2