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

$\mathbf {E} [X]=\sum _{n=0}^{\infty }(n+1)q^{n}p$Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{p}{(1-q)^2}= \frac{1}{p}}

Similarly, the variance can be calculated:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Var(X) = \frac{1-p}{p^2}}