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.
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
where
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
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
-
and the
variance is
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
Two other important distributions arise as approximations of binomial distributions:
Binomial PDF and Normal approximation for n=6 and p=0.5
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.