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单词 GeometricDistribution
释义

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

f(x)=q(x-1)p.

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

F(x)=kxq(k-1)p.

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

G(t)=k=1etkq(k-1)p=petk=0(etq)k.

The last expression can be simplified as

G(t)=pet1-etq.

The first derivativeMathworldPlanetmath is

G(t)=etp(1-etq)2

so the mean is

μ=E[X]=G(0)=1p.

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

E[X2]=G′′(0)=2-pp2

and therefore the variance is

σ2=E[X2]-E[X]2=G′′(0)-G(0)2=qp2.
Titlegeometric distribution
Canonical nameGeometricDistribution
Date of creation2013-03-22 13:03:07
Last modified on2013-03-22 13:03:07
OwnerMathprof (13753)
Last modified byMathprof (13753)
Numerical id14
AuthorMathprof (13753)
Entry typeDefinition
Classificationmsc 60E05
Synonymgeometric random variable
Related topicRandomVariable
Related topicDensityFunction
Related topicDistributionFunction
Related topicMean
Related topicVariance
Related topicBernoulliDistribution
Related topicArithmeticMean
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