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Binomial Distribution

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Probability Distributions -> Discrete Distributions -> Binomial Distribution

Bernoulli Distribution

The Bernoulli distribution is a discrete probability distribution with only two possible values for the random variable.

Binomial Distribution

The binomial distribution models the total number of successes in repeated trials from an infinite population under certain conditions.

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Overview

Parameters

Probability Density Function

Cumulative Distribution Function

Descriptive Statistics

Example

Related Distributions

References

See Also

Related Topics

Main Content

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 object BinomialDistribution by 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 the Distribution Fitter app. 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') and parameters.

Parameters

The binomial distribution uses the following parameters.

Parameter Description Support

N Number of trials Positive integer

p Probability of success in a single trial 0≤p≤1

The sum of two binomial random variables that both have the same parameter p is also a binomial random variable with N equal 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)=(

N

x

)p

x

(1−p)

N−x

 ; x=0,1,2,...,N ,

where x is the number of successes in N trials of a Bernoulli process with the probability of success p. The result is the probability of exactly x successes in N trials. For discrete distributions, the pdf is also known as the probability mass function (pmf).

For an example, see Compute Binomial Distribution pdf.

Cumulative Distribution Function

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

F(x∣N,p)=

x

∑

i=0

(

N

i

)p

i

(1−p)

N−i

 ; x=0,1,2,...,N ,

where x is the number of successes in N trials of a Bernoulli process with the probability of success p. The result is the probability of at most x successes in N trials.

For an example, see Compute Binomial Distribution cdf.

Descriptive Statistics

The mean of the binomial distribution is Np.

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

Example

Fit Binomial Distribution to Data

Generate a binomial random number that counts the number of successes in 100 trials with the probability of success 0.9 in each trial.

x = binornd(100,0.9)

x = 85

Fit a binomial distribution to data using fitdist.

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

pd =

BinomialDistribution

Binomial distribution

N = 100

p = 0.85 [0.764692, 0.913546]

fitdist returns a BinomialDistribution object. The interval next to p is the 95% confidence interval estimating p.

Estimate the parameter p using the distribution functions.

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

phat = 0.8500

pci = 1×2

0.7647    0.9135

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

phat2 = 0.8500

pci2 = 2×1

0.7647
0.9135

Compute Binomial Distribution pdf

Compute the pdf of the binomial distribution with 10 trials and the probability of success 0.5.

x = 0:10;

y = binopdf(x,10,0.5);

Plot the pdf with bars of width 1.

figure

bar(x,y,1)

xlabel('Observation')

ylabel('Probability')

Compute Binomial Distribution cdf

Compute the cdf of the binomial distribution with 10 trials and the probability of success 0.5.

x = 0:10;

y = binocdf(x,10,0.5);

Plot the cdf.

figure

stairs(x,y)

xlabel('Observation')

ylabel('Cumulative Probability')

Compare Binomial and Normal Distribution pdfs

When N is large, the binomial distribution with parameters N and p can be approximated by the normal distribution with mean Np and variance Np*(1–p) provided that p is not too large or too small.

Compute the pdf of the binomial distribution counting the number of successes in 50 trials with the probability 0.6 in 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 = Np;

sigma = sqrt(Np*(1-p));

x2 = 0:0.1:N;

y2 = normpdf(x2,mu,sigma