Binomial distribution

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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.

Occurrence

A typical example is the following: assume 5% of the population is green-eyed. You pick 500 people randomly. How likely is it that you get 30 or more green-eyed people? The number of green-eyed people you pick is a random variable X which follows a binomial distribution with n = 500 and p = 0.05 (when picking the people with replacement). We are interested in the probability Pr[X ≥ 30].

Specification

Probability mass function

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 the probability mass function:

[f(k;n,p)=p^k(1-p)^\,]

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

[=\frac]

is the binomial coefficient "n choose k" (also denoted C(n, k) or nCk), hence 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.

In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as

[f(k;n,p)=f(n-k;n,1-p).\,]

So, one must look to a different k and a different p (the binomial is not symmetrical in general).

Cumulative distribution function

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

[ F(k;n,p) = \Pr(X \le k) = I_(n-k, k+1) \!]

provided k is an integer and 0 ≤ k ≤ n. If x is not necessarily an integer or not necessarily positive, one can express it thus:

[F(x;n,p) = \Pr(X \le x) = \sum_^ p^j(1-p)^]

where [\lfloor x\rfloor] is the greatest integer less than or equal to x.

For [k \le np], upper bounds for the lower tail of the distribution function can be derived. In particular, Hoeffding's inequality yields the bound

[ F(k;n,p) \leq \exp\left(-2 \frac\right), \!]

and Chernoff's inequality can be used to derive the bound

[ F(k;n,p) \leq \exp\left(-\frac \frac\right). \!]

Mean, standard deviation, and mode

If X ~ B(n, p) (that is, X is a binomially distributed random variate), then the expected value of X is

[E[X]=np\,]

and the variance is

[\mbox(X)=np(1-p).\,]

This fact is easily proven as follows. Suppose first that we have exactly one Bernoulli trial. We have two possible outcomes, 1 and 0, with the first having probability p and the second having probability 1 − p; the mean for this trial is given by μ = p. Using the definition of variance, we have

[\sigma^2= \left(1 - p\right)^2p + (-p)^2(1 - p) = p(1-p).]

Now suppose that we want the variance for n such trials (i.e. for the general binomial distribution). Since the trials are independent, we may add the variances for each trial, giving

[\sigma^2_n = \sum_^n \sigma^2 = np(1 - p). \quad \Box]

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.

Relations to other distributions

• 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

:[X+Y \sim B(n+m, p).\,]

Two other important distributions arise as approximations of binomial distributions:

• If n is large enough, the skew of the distribution is not too great, and a suitable continuity correction is used, then an excellent approximation to B(n, p) is given by the normal distribution

:[ N(np, np(1-p)).\,]

Various rules of thumb may be used to decide whether n is large enough. One rule is that both np and n(1 − p) must be greater than 5. However, the specific number varies from source to source, and depends on how good an approximation one wants; some sources give 10. Another commonly used rule holds that the above normal approximation is appropriate only if

:[\mu \pm 3 \sigma = np \pm 3 \sqrt \in [0,n].]

The following is an example of applying a continuity correction: Suppose one wishes to calculate Pr(X ≤ 8) for a binomial random variable X. If Y has a distribution given by the normal approximation, then Pr(X ≤ 8) is approximated by Pr(Y ≤ 8.5). The addition of 0.5 is the continuity correction. Warning: The normal approximation gives inaccurate results unless a continuity correction is used.

This approximation is a huge time-saver (exact calculations with large n are very onerous); 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.

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(n, p) distribution approaches the Poisson distribution with expected value λ.

• As n approaches ∞ while p remains fixed, the distribution of

:[}]

approaches the normal distribution with expected value 0 and variance 1.

References

• Luc Devroye, Non-Uniform Random Variate Generation, New York: Springer-Verlag, 1986. See especially [Chapter X, Discrete Univariate Distributions].
• Voratas Kachitvichyanukul and Bruce W. Schmeiser, Binomial random variate generation, Communications of the ACM 31(2):216–222, February 1988. DOI:[10.1145/42372.42381]

See also

• Beta distribution
• Multinomial distribution
• Negative binomial distribution
• Poisson distribution
• Hypergeometric distribution

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External links

• [Binomial Probability Distribution Calculator]

 

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