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


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

The corresponding distribution function is


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


which simplifies to


Differentiating gives us


and therefore the mean is


Now for the variance we need the second derivative


so we get


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


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.

Title binomial distribution
Canonical name BinomialDistribution
Date of creation 2013-03-22 13:03:01
Last modified on 2013-03-22 13:03:01
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 17
Author Mathprof (13753)
Entry type Definition
Classification msc 60E05
Synonym Bernoulli distribution
Synonym binomial random variableMathworldPlanetmath
Synonym binomial probability function
Synonym Bernoulli random variable
Related topic BinomialCoefficient
Related topic BinomialTheorem
Related topic BernoulliRandomVariable