Binomial distribution

In mathematics, the binomial distribution is a discrete probability distribution which describes the number of successes in a sequence of n independent yes/no experiments, each of which yielding success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. 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 are 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

$P[X=k]={n\choose k}p^k(1-p)^{n-k}\quad\mbox{for}\ k=0,1,2,\dots,n$

where

${n\choose k}=\frac{n!}{k!(n-k)!}$

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 (p^k) and n − k failures ((1 − p)^{n − k}). 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:

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

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

E[X] = np

and the variance is

var(X) = np(1 - p).

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

B(n + m,p).

Two other important distributions arise as approximations of binomial distributions:

Binomial PDF and Normal approximation for n=6 and p=0.5.

If both np and n(1 − p) are greater than 5 or so, then an excellent approximation (provided a suitable continuity correction is used) to B(n, p) is given by the normal distribution

N(np,np(1 - p)).

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.

Binomial distribution

See also