Difference between revisions of "Geometric distribution"
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{0,1,2,…}[[category:Probability and Statistics]] | {0,1,2,…}[[category:Probability and Statistics]] | ||
| − | == Mean | + | == Mean and Variance == |
The [[mean]] for a [[random variable]] ''X'' following a geometric distribution | The [[mean]] for a [[random variable]] ''X'' following a geometric distribution | ||
with a probability of success ''p'' (and ''q'' = 1 - ''p'') is | with a probability of success ''p'' (and ''q'' = 1 - ''p'') is | ||
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<math>\mathbf{E}[X] = \sum_{n=0}^{\infty} (n+1) q^n | <math>\mathbf{E}[X] = \sum_{n=0}^{\infty} (n+1) q^n | ||
p</math><math>=\frac{p}{(1-q)^2}= \frac{1}{p}</math> | p</math><math>=\frac{p}{(1-q)^2}= \frac{1}{p}</math> | ||
| + | |||
| + | Similarly, the variance can be calculated: | ||
| + | |||
| + | <math>Var(X) = \frac{1-p}{p^2}</math> | ||
Revision as of 16:46, August 3, 2010
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
![{\displaystyle \mathbf {E} [X]=\sum _{n=0}^{\infty }(n+1)q^{n}p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b68c5fe4a407b29c0a3d12a08e6bae0fa9259358)
Similarly, the variance can be calculated:
