Binomial Distribution

The binomial distribution, also known as the Bernoulli distribution, models the number of successes in a fixed number of Bernoulli trials. A Bernoulli trial is an experiment with two possible outcomes, labeled 'success' and 'failure,' where the probability of success has a fixed value for all trials.

The binomial distribution has two parameters: the number of trials n and the probability of success p in each trial. The probability mass function (PMF) is given by:

$$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$$

The cumulative distribution function (CDF) is:

$$F(k;n,p)=\sum _{i=0}^k(\genfrac{}{}{0ex}{}{n}{i})p^i(1-p{)}^{n-i}$$

The domain of the binomial distribution is $k\in \{0,1,2,\dots ,n\}$. The parameters must satisfy $n\in \mathbb{N}$ and $0\le p\le 1$.

The binomial distribution arises whenever underlying events have two possible outcomes, and the probability of each outcome occurring remains constant. Common applications include:

• The number of times heads or tails is obtained in a fixed number of coin tosses.

• The number of defective elements found in a random sample of fixed size from a stable production process.

• Estimating the size of an animal population by marking individuals and releasing them back into the wild.

The binomial distribution has several important statistical properties:

The binomial distribution is closely related to several other distributions:

• The Bernoulli Distribution is a special case of the binomial distribution with $n=1$.

• The Geometric Distribution models the number of failures before the first success in a series of Bernoulli trials.

• The Negative Binomial Distribution models the number of failures before the nth success in a series of Bernoulli trials.

The binomial distribution is implemented by the BinomialDistribution class. It has two constructors. The first constructor takes one argument: the number of trials. The probability of success is assumed to be 0.5. The following constructs a binomial distribution for 10 trials:

var binomial1 = new BinomialDistribution(10);

Dim binomial1 = New BinomialDistribution(10)

No code example is currently available or this language may not be supported.

let binomial1 = BinomialDistribution(10)

The second constructor takes two arguments. The first argument is once again the number of trials. The second argument is the probability of success of a trial. The probability must be between 0 and 1. The following constructs a binomial distribution with 10 trials and probability of success 0.4:

var binomial2 = new BinomialDistribution(10, 0.4);

Dim binomial2 = New BinomialDistribution(10, 0.4)

No code example is currently available or this language may not be supported.

let binomial2 = BinomialDistribution(10, 0.4)

The BinomialDistribution class has two specific properties: NumberOfTrials returns the number of trials. ProbabilityOfSuccess returns the probability of success of a trial.

BinomialDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. It has two overloads, corresponding to each of the two constructors.

var random = new Pcg32();
int sample1 = BinomialDistribution.Sample(random, 10);
int sample2 = BinomialDistribution.Sample(random, 10, 0.4);

Dim random = New Pcg32()
Dim sample1 = BinomialDistribution.Sample(random, 10)
Dim sample2 = BinomialDistribution.Sample(random, 10, 0.4)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample1 = BinomialDistribution.Sample(random, 10)
let sample2 = BinomialDistribution.Sample(random, 10, 0.4)

The above example uses the Pcg32 class to generate uniform random numbers.

For details of the properties and methods common to all discrete probability distribution classes, see the topic on Discrete Probability Distributions.

• "Introduction to Probability Models" by Sheldon M. Ross.

• "Probability and Statistics" by Morris H. DeGroot and Mark J. Schervish.

Other Resources