Binomial Distribution

Overview

The binomial distribution is a two-parameter family of curves. The binomial distribution is used to model the total number of successes in a fixed number of independent trials that have the same probability of success, such as modeling the probability of a given number of heads in ten flips of a fair coin.

Statistics and Machine Learning Toolbox™ offers several ways to work with the binomial distribution.

Create a probability distribution objectBinomialDistributionby fitting a probability distribution to sample data (fitdist) or by specifying parameter values (makedist). Then, use object functions to evaluate the distribution, generate random numbers, and so on.
Work with the binomial distribution interactively by using theDistribution Fitterapp. You can export an object from the app and use the object functions.
Use distribution-specific functions (binocdf,binopdf,binoinv,binostat,binofit,binornd) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple binomial distributions.
Use generic distribution functions (cdf,icdf,pdf,random) with a specified distribution name ('Binomial')和参数。

Parameters

The binomial distribution uses the following parameters.

Parameter	Description	金宝app
N	Number of trials	Positive integer
p	Probability of success in a single trial	$0 \leq p \leq 1$

The sum of two binomial random variables that both have the same parameterpis also a binomial random variable withNequal to the sum of the number of trials.

Probability Density Function

The probability density function (pdf) of the binomial distribution is

$f (x | N, p) = (\begin{matrix} N \\ x \end{matrix}) p^{x} {(1 - p)}^{N - x}; x = 0, 1, 2, ..., N,$

wherexis the number of successes inNtrials of a Bernoulli process with the probability of successp. The result is the probability of exactlyxsuccesses inNtrials. For discrete distributions, the pdf is also known as the probability mass function (pmf).

For an example, seeCompute Binomial Distribution pdf.

Cumulative Distribution Function

The cumulative distribution function (cdf) of the binomial distribution is

$F (x | N, p) = \sum_{i = 0}^{x} (\begin{matrix} N \\ i \end{matrix}) p^{i} {(1 - p)}^{N - i}; x = 0, 1, 2, ..., N,$

wherexis the number of successes inNtrials of a Bernoulli process with the probability of successp. The result is the probability of at mostxsuccesses inNtrials.

For an example, seeCompute Binomial Distribution cdf.

Descriptive Statistics

The mean of the binomial distribution isNp.

The variance of the binomial distribution isNp(1 –p).

Example

Fit Binomial Distribution to Data

Open Live Script

Generate a binomial random number that counts the number of successes in100trials with the probability of success0.9in each trial.

x = binornd(100,0.9)

x = 85

Fit a binomial distribution to data usingfitdist.

pd = fitdist(x,'Binomial','NTrials',100)

pd = BinomialDistribution Binomial distribution N = 100 p = 0.85 [0.764692, 0.913546]

fitdist返回一个BinomialDistributionobject. The interval next topis the 95% confidence interval estimatingp.

Estimate the parameterpusing the distribution functions.

[phat,pci] = binofit(x,100)% Distribution-specific function

phat = 0.8500

pci =1×20.7647 0.9135

[phat2,pci2] = mle(x,'distribution','Binomial',"NTrials",100)% Generic distribution function

phat2 = 0.8500

pci2 =2×10.7647 0.9135

Compute Binomial Distribution pdf

Open Live Script

Compute the pdf of the binomial distribution with10trials and the probability of success0.5.

x = 0:10; y = binopdf(x,10,0.5);

Plot the pdf with bars of width1.

figure bar(x,y,1) xlabel('Observation') ylabel('Probability')

Figure contains an axes object. The axes object with xlabel Observation, ylabel Probability contains an object of type bar.

Compute Binomial Distribution cdf

Open Live Script

Compute the cdf of the binomial distribution with10trials and the probability of success0.5.

x = 0:10; y = binocdf(x,10,0.5);

Plot the cdf.

figure stairs(x,y) xlabel('Observation') ylabel('Cumulative Probability')

Figure contains an axes object. The axes object with xlabel Observation, ylabel Cumulative Probability contains an object of type stair.

Compare Binomial and Normal Distribution pdfs

Open Live Script

WhenNis large, the binomial distribution with parametersNandpcan be approximated by the normal distribution with meanN*pand varianceN*p*(1–p)provided thatpis not too large or too small.

Compute the pdf of the binomial distribution counting the number of successes in50trials with the probability0.6in a single trial .

N = 50; p = 0.6; x1 = 0:N; y1 = binopdf(x1,N,p);

Compute the pdf of the corresponding normal distribution.

mu = N*p; sigma = sqrt(N*p*(1-p)); x2 = 0:0.1:N; y2 = normpdf(x2,mu,sigma);

情节pdf文档在同一轴。

figure bar(x1,y1,1) holdonplot(x2,y2,'LineWidth',2) xlabel('Observation') ylabel('Probability') title('Binomial and Normal pdfs') legend('Binomial Distribution','Normal Distribution','location','northwest') holdoff

Figure contains an axes object. The axes object with title Binomial and Normal pdfs, xlabel Observation, ylabel Probability contains 2 objects of type bar, line. These objects represent Binomial Distribution, Normal Distribution.

The pdf of the normal distribution closely approximates the pdf of the binomial distribution.

Compare Binomial and Poisson Distribution pdfs

Open Live Script

Whenpis small, the binomial distribution with parametersNandpcan be approximated by the Poisson distribution with meanN*p, provided thatN*pis also small.

Compute the pdf of the binomial distribution counting the number of successes in20trials with the probability of success0.05in a single trial.

N = 20; p = 0.05; x = 0:N; y1 = binopdf(x,N,p);

Compute the pdf of the corresponding Poisson distribution.

mu = N*p; y2 = poisspdf(x,mu);

情节pdf文档在同一轴。

figure bar(x,[y1; y2]) xlabel('Observation') ylabel('Probability') title('Binomial and Poisson pdfs') legend('Binomial Distribution','Poisson Distribution','location','northeast')

Figure contains an axes object. The axes object with title Binomial and Poisson pdfs, xlabel Observation, ylabel Probability contains 2 objects of type bar. These objects represent Binomial Distribution, Poisson Distribution.

The pdf of the Poisson distribution closely approximates the pdf of the binomial distribution.

Related Distributions

Bernoulli Distribution— The Bernoulli distribution is a one-parameter discrete distribution that models the success of a single trial, and occurs as a binomial distribution withN= 1.
Multinomial Distribution— The multinomial distribution is a discrete distribution that generalizes the binomial distribution when each trial has more than two possible outcomes.
Normal Distribution— The normal distribution is a two-parameter continuous distribution that has parametersμ(mean) andσ(standard deviation). AsNincreases, the binomial distribution can be approximated by a normal distribution withµ=Npandσ²=Np(1 –p). SeeCompare Binomial and Normal Distribution pdfs.
Poisson Distribution— The Poisson distribution is a one-parameter discrete distribution that takes nonnegative integer values. The parameterλis both the mean and the variance of the distribution. The Poisson distribution is the limiting case of a binomial distribution whereNapproaches infinity andpgoes to zero whileNp=λ. SeeCompare Binomial and Poisson Distribution pdfs.

References

[1] Abramowitz, Milton, and Irene A. Stegun, eds.Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. 9. Dover print.; [Nachdr. der Ausg. von 1972]. Dover Books on Mathematics. New York, NY: Dover Publ, 2013.

[2] Evans, Merran, Nicholas Hastings, and Brian Peacock.Statistical Distributions. 2nd ed. New York: J. Wiley, 1993.

[3] Loader, Catherine.Fast and Accurate Computation of Binomial Probabilities. July 9, 2000.