# Binomial distribution

See binomial (disambiguation) for a list of other topics using that name.

In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. In fact, when n = 1, then the binomial distribution is the Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

A typical example is the following: assume 5% of the population is HIV-positive. You pick 500 people randomly. How likely is it that you get 30 or more HIV-positives? The number of HIV-positives you pick is a random variable X which follows a binomial distribution with n = 500 and p = .05. We are interested in the probability Pr[X ≥ 30].

In general, if the random variable X follows the binomial distribution with parameters n and p, we write X ~ B(n, p). The probability of getting exactly k successes is given by

[itex]P[X=k]={n\choose k}p^k(1-p)^{n-k}\![itex]

for [itex]k=0,1,2,\dots,n[itex] and where

[itex]{n\choose k}=\frac{n!}{k!(n-k)!}[itex]

is the binomial coefficient "n choose k" (also denoted C(n, k)), whence the name of the distribution. The formula can be understood as follows: we want k successes (pk) and nk failures ((1 − p)nk). However, the k successes can occur anywhere among the n trials, and there are C(n, k) different ways of distributing k successes in a sequence of n trials.

The cumulative distribution function can be expressed in terms of the regularized incomplete beta function, as follows:

[itex] F(k) = I_{1-p}(n-k, k+1) [itex].

If X ~ B(n, p), then the expected value of X is

[itex]E[X]=np[itex]

and the variance is

[itex]\mbox{var}(X)=np(1-p).[itex]

The most likely value or mode of X is given by the largest integer less than or equal to (n+1)p; if m = (n+1)p is itself an integer, then m − 1 and m are both modes.

If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables, then X + Y is again a binomial variable; its distribution is

[itex]B(n+m, p).[itex]

Two other important distributions arise as approximations of binomial distributions:

Missing image
BinDistApprox_large.png
Binomial PDF and Normal approximation for n=6 and p=0.5.
[itex] N(np, np(1-p)).[itex]
This approximation is a huge time-saver; historically, it was the first use of the normal distribution, introduced in Abraham de Moivre's book The Doctrine of Chances in 1733. Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a sum of n independent, identically distributed 0-1 indicator variables. Warning: this approximation gives inaccurate results unless a continuity correction is used. Note: that the picture gives the normal and binomial probability density functions (PDF) and not the cumulative distribution functions.
For example, suppose you randomly sample n people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If you sampled groups of n people repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation σ = (p(1 − p)/n)1/2. Large sample sizes n are good because the standard deviation gets smaller, which allows a more precise estimate of the unknown parameter p.
• If n is large and p is small, so that np is of moderate size, then the Poisson distribution with parameter λ = np is a good approximation to B(n, p).

The formula for Bézier curves was inspired by the binomial distribution.

## Limits of binomial distributions

• As n approaches ∞ and p approaches 0 while np remains fixed at λ > 0 or at least np approaches λ > 0, then the Binomial(np) distribution approaches the Poisson distribution with expected value λ.
• As n approaches ∞ while p remains fixed, the distribution of
[itex]{X-np \over \sqrt{np(1-p)\ }}[itex]
approaches the normal distribution with expected value 0 and variance 1.

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