represents a binomial distribution with n trials and success probability p.
- The probability for value in a binomial distribution is for integers from 0 to n. »
- BinomialDistribution allows n to be any non-negative integer.
- BinomialDistribution allows n and p to be dimensionless quantities. »
- BinomialDistribution can be used with such functions as Mean, CDF, and RandomVariate. »
Background & Context
- BinomialDistribution[n,p] represents a discrete statistical distribution defined at integer values and parametrized by a non-negative real number p, . The binomial distribution has a discrete probability density function (PDF) that is unimodal, with its peak occurring at the mean . The parameters n and p determine the height, horizontal location, and skewness of the PDF.
- The study of the binomial distribution dates back to the early eighteenth century to the work of James Bernoulli, thus making it one of the oldest distributions to be studied. The binomial distribution is designed to model the action of flipping n (fair or unfair) coins that are independent and equal and are sampled independently and sequentially with replacement. Traditionally, p is thought of as the probability with which the experiment "succeeds", whereas is the probability of "failure". In the coin flip analogy, the value corresponds to flipping a fair coin.
- Despite being defined very simply, the binomial distribution serves as the basis for a number of more complicated mathematical concepts. For example, the binomial distribution can be thought of as the distribution of successes resulting from a finite n-stage Bernoulli process of having probability of success p (i.e. a discrete-time stochastic process consisting of a finite sequence of random variables, each of which is independent and identically distributed (i.i.d.) according to BernoulliDistribution[p]). Similarly, the binomial distribution is the slice distribution (SliceDistribution) of a binomial process (BinomialProcess), a discrete-time, discrete-state stochastic process consisting of a finite sequence of i.i.d. random variables following a binomial distribution, the time between which follows a geometric distribution (GeometricDistribution). Moreover, many real-world scenarios can be modeled as binomial processes, e.g. the probability of rolling a particular value among n (fair or unfair) dice. Surprising connections between the binomial distribution have also been discovered in the study of emigration patterns and in certain queueing models.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a binomial distribution. Distributed[x,BinomialDistribution[n,p]], written more concisely as xBinomialDistribution[n,p], can be used to assert that a random variable x is distributed according to a binomial distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions may be given using PDF[BinomialDistribution[n,p],x] and CDF[BinomialDistribution[n,p],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. These quantities can be visualized using DiscretePlot.
- DistributionFitTest can be used to test if a given dataset is consistent with a binomial distribution, EstimatedDistribution to estimate a binomial parametric distribution from given data, and FindDistributionParameters to fit data to a binomial distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic binomial distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic binomial distribution.
- TransformedDistribution can be used to represent a transformed binomial distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a binomial distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving binomial distributions.
- BinomialDistribution is related to a number of other statistical distributions. For example, BinomialDistribution[1,p] is precisely the same as BernoulliDistribution[p] on the values and , and the sum of n independent variables distributed according to BernoulliDistribution[p] is distributed according to BinomialDistribution[n,p]. Similarly, BinomialDistribution[t,p] has the same PDF as BinomialProcess[p][t], a property due to the fact that BinomialDistribution[t,p] is precisely SliceDistribution[BinomialProcess[p],t]. BinomialDistribution[n,p] is also the limiting distribution for several distributions. In particular, BinomialDistribution[n,p] converges to NormalDistribution[μ,σ] for fixed p as n tends to Infinity where and and converges to a discretized PoissonDistribution[p] as n tends to Infinity and p tends to 0, while HypergeometricDistribution[n,p] limits to BinomialDistribution[n,p] as p tends to Infinity. BinomialDistribution is a two-variable case of MultinomialDistribution, is a constituent piece of BetaBinomialDistribution, and has a natural relationship with NegativeBinomialDistribution.
Examplesopen allclose all
Basic Examples (3)
The limiting value is the value of kurtosis of the standard NormalDistribution:
The number of heads in n flips with a fair coin can be modeled with BinomialDistribution:
A system uses triple redundancy with three microprocessors and is designed to operate as long as one processor is still functional. The probability that a microprocessor is still functional after seconds is . Find the probability that the system is still operating after seconds:
Gary Kasparov, chess champion, plays in a tournament simultaneously against 100 amateurs. It has been estimated that he loses about 1% of such games. Find the probability of losing 0, 2, 5, and 10 games:
A packet consisting of a string of n symbols is transmitted over a noisy channel. Each symbol has probability of incorrect transmission. Find n such that the probability of incorrect packet transmission is less than :
Properties & Relations (9)
BinomialDistribution[n,p] converges to a normal distribution as :
The sum of n independent variables with BernoulliDistribution is binomial distributed: