Binomial distribution

The binomial distribution is a discrete probability distribution which describes the number of successes in a sequence of n independent experiments, each of which yielding success with probability p. Such a success/failure experiment is also called a Bernoulli experiment.

A typical example is the following: 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

Pr[X = k] = C(n, k) p^k (1-p)^n-k     for k = 0, 1, 2, ..., n

Here, C(n, k) denotes the binomial coefficient of n and 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.

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:

• If both np and n(1-p) are greater than 5 or so, then an excellent approximation 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. 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.
  • 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).

pictures of these approximations would be nice.

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