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

$\mathbf {E} [X]=\sum _{n=0}^{\infty }(n+1)q^{n}p$$={\frac {p}{(1-q)^{2}}}={\frac {1}{p}}$

The variance can be calculated similarly:

$Var(X)={\frac {1-p}{p^{2}}}$.

Probability-generating function

The probability-generating function $G_{X}(z)=\sum _{n=0}^{\infty }z^{n}\cdot \mathbf {P} [X=n]$ is:

$G_{X}(z)={\frac {z\,p}{1-z(1-p)}}$.