Binomial distribution

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Story

We perform \(N\) Bernoulli trials, each with probability \(\theta\) of success. The number of successes, \(n\), is Binomially distributed.

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Example

Distribution of plasmids between daughter cells in cell division. Each of the \(N\) plasmids as a chance \(\theta\) of being in daughter cell 1 (“success”). The number of plasmids, \(n\), in daughter cell 1 is Binomially distributed.

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Parameters

There are two parameters: the number of Bernoulli trials, \(N\), and the probability \(\theta\) of success for each trial.

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Support

The Binomial distribution is supported on the set of nonnegative integers less than or equal to \(N\).

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Probability mass function

\[\begin{split}\begin{align} f(n;N,\theta) = \begin{pmatrix} N \\ n \end{pmatrix} \theta^n (1-\theta)^{N-n}. \end{align}\end{split}\]

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Cumulative distribution function

The CDF evaluated at integers \(n\) with \(0 \le n \le N\) is

\[\begin{align} F(n;N,\theta) = I_{1-\theta}(N - n, n + 1), \end{align}\]

a regularized incomplete beta function.

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Moments

Mean: \(N\theta\)

Variance: \(N\theta(1-\theta)\)

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Usage

Package	Syntax
NumPy	rng.binomial(N, theta)
SciPy	scipy.stats.binom(N, theta)
Distributions.jl	Binomial(N, theta)
Stan	binomial(N, theta)

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Related distributions

• The Bernoulli distribution is a special case of the Binomial distribution where \(N=1\).

• In the limit of \(N\to\infty\) and \(\theta\to 0\) such that the quantity \(N\theta\) is fixed, the Binomial distribution becomes a Poisson distribution with parameter \(N\theta\).

• The Binomial distribution is further connected to the Poisson distribution. Consider two Poisson processes with arrival rates \(\lambda_1\) and \(\lambda_2\) such that the number of arrivals of the respective processes are \(n_1\sim \text{Poisson}(\lambda_1)\) and \(n_2\sim \text{Poisson}(\lambda_2)\). Let \(\theta = \lambda_1 / (\lambda_1 + \lambda_2)\) and let \(N = n_1 + n_2\) be the total number of arrivals of both Poisson processes. Then, \(N \sim \text{Poisson}(\lambda_1 + \lambda_2)\) and \(n_1 \sim \text{Binom}(N, \theta)\).

• The Binomial distribution is a limit of the Hypergeometric distribution. Considering the Hypergeometric distribution parametrized by \(N\), \(a\), and \(b\), and taking the limit of \(a+b\to\infty\) such that \(a/(a+b)\) is fixed, we get a Binomial distribution with parameters \(N=N\) and \(\theta = a/(a+b)\).

• The Binomial distribution is a special case of the Multinomial distribution when \(K = 2\).

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PMF and CDF plots

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Links

• Wikipedia

• Numpy

• Scipy

• Distributions.jl

• Stan