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单词 BinomialDistribution
释义

binomial distribution


Consider an experiment with two possible outcomes (success and failure), which happen randomly. Let p be the probability of success. If the experiment is repeated n times, the probability of having exactly x successes is

f(x)=(nx)px(1-p)(n-x).

The distribution functionMathworldPlanetmath determined by the probability functionDlmfDlmfPlanetmath f(x) is called a Bernoulli distributionMathworldPlanetmath or binomial distribution.

Here are some plots for f(x) with n=20 and p=0.3, p=0.5.

The corresponding distribution function is

F(x)=kx(nk)pkqn-k

where q=1-p. Notice that if we calculate F(n) we get the binomial expansion for (p+q)n, and this is the reason for the distributionDlmfPlanetmath being called binomial.

We will use the moment generating function to calculate the mean and varianceMathworldPlanetmath for the distribution. The mentioned functionMathworldPlanetmath is

G(t)=x=0netx(nx)pxqn-x

which simplifies to

G(t)=(pet+q)n.

Differentiating gives us

G(t)=n(pet+q)n-1pet

and therefore the mean is

μ=E[X]=G(0)=np.

Now for the variance we need the second derivative

G′′(t)=n(n-1)(pet+q)n-2+n(pet+q)n-1pet

so we get

E[X2]=G′′(0)=n(n-1)p2+np

and finally the variance (recall q=1-p):

σ2=E[X2]-E[X]2=npq.

For large values of n, the binomial coefficientsDlmfDlmfMathworldPlanetmath are hard to compute, however in this cases we can use either the Poisson distributionMathworldPlanetmath or the normal distributionMathworldPlanetmath to approximate the probabilities.

Titlebinomial distribution
Canonical nameBinomialDistribution
Date of creation2013-03-22 13:03:01
Last modified on2013-03-22 13:03:01
OwnerMathprof (13753)
Last modified byMathprof (13753)
Numerical id17
AuthorMathprof (13753)
Entry typeDefinition
Classificationmsc 60E05
SynonymBernoulli distribution
Synonymbinomial random variableMathworldPlanetmath
Synonymbinomial probability function
SynonymBernoulli random variable
Related topicBinomialCoefficient
Related topicBinomialTheorem
Related topicBernoulliRandomVariable
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更新时间:2025/5/4 14:43:21