Geometric distribution

The geometric distribution is a discrete distribution which describes the number of repetitions of a Bernoulli experiment to get a success. Therefore, its support are the positive integers, {1,2,3,…}

Alternative definition

Some authors describe the geometric distribution as the number of repetitions until the first success, its support being the non-negative integers {0,1,2,…}

Mean and Variance

The mean for a random variable X following a geometric distribution with a probability of success p (and q = 1 - p) is

<math>\mathbf{E}[X] = \sum_{n=0}^{\infty} (n+1) q^n p</math><math>=\frac{p}{(1-q)^2}= \frac{1}{p}</math>

The variance can be calculated similarly:

<math>Var(X) = \frac{1-p}{p^2}</math>.

Probability-generating function

The probability-generating function <math>G_X(z) = \sum_{n=0}^\infty z^n \cdot \mathbf{P}[X=n]</math> is:

<math>G_X(z) = \frac{z\,p}{1-z(1-p)}</math>.