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The '''geometric distribution''' is a discrete distribution which describes the
The '''geometric distribution''' is a discrete distribution which describes the
number of repetitions of a [[Bernoulli experiment]] to get a success. Therefore
number of repetitions of a [[Bernoulli experiment]] to get a success. Therefore,
its support are the positive integers, {1,2,3,…}
its support are the positive integers, {1,2,3,…}
Some authors describe the geometric distribution as the number of repetitions
Some authors describe the geometric distribution as the number of repetitions
until the first success, its support being the non-negative integers
until the first success, its support being the non-negative integers
{0,1,2,…}[[category:Probability and Statistics]]
{0,1,2,…}
== Mean and Variance ==
== Mean and Variance ==
p</math><math>=\frac{p}{(1-q)^2}= \frac{1}{p}</math>
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>
<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>.
[[Category:Probability and Statistics]]