geometric distribution

Suppose that a random experiment has two possible outcomes, success with probability p and failure with probability q=1-p. The experiment is repeated until a success happens. The number of trials before the success is a random variableMathworldPlanetmath X with density function


The distribution functionMathworldPlanetmath determined by f(x) is called a geometric distributionMathworldPlanetmathPlanetmath with parameter p and it is given by


The picture shows the graph for f(x) with p=1/4. Notice the quick decreasing. An interpretationMathworldPlanetmathPlanetmath is that a long run of failures is very unlikely.

We can use the moment generating function method in order to get the mean and varianceMathworldPlanetmath. This function is


The last expression can be simplified as


The first derivativeMathworldPlanetmath is


so the mean is


In order to find the variance, we use the second derivative and thus


and therefore the variance is

Title geometric distribution
Canonical name GeometricDistribution
Date of creation 2013-03-22 13:03:07
Last modified on 2013-03-22 13:03:07
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 14
Author Mathprof (13753)
Entry type Definition
Classification msc 60E05
Synonym geometric random variable
Related topic RandomVariable
Related topic DensityFunction
Related topic DistributionFunction
Related topic Mean
Related topic Variance
Related topic BernoulliDistribution
Related topic ArithmeticMean