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1 Binomial distribution From Wikipedia, the free encyclopedia See also: Negative binomial distribution 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; when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution is a good approximation, and widely used. Contents 1 Specification 1.1 Probability mass function 1.2 Cumulative distribution function 2 Mean and variance 3 Mode and median 4 Covariance between two binomials 5 Relationship to other Probability mass function Cumulative distribution function Notation B(n, p) Parameters n N 0 number of trials p [0,1] success probability in each trial Support k { 0,, n } number of successes PMF CDF Mean np Median np or np Mode (n + 1)p or (n + 1)p 1 Variance np(1 p) Skewness 1/9

2 distributions 5.1 Sums of binomials 5.2 Conditional binomials 5.3 Bernoulli distribution 5.4 Poisson binomial distribution 5.5 Normal approximation 5.6 Poisson approximation 5.7 Limiting distributions 6 Generating binomial random variates 7 See also 8 References Ex. kurtosis Entropy MGF CF PGF Specification Probability mass function Binomial distribution for (blue), (green) and (red) In general, if the random variable K follows the binomial distribution with parameters n and p, we write K ~ B(n, p). The probability of getting exactly k successes in n trials is given by the probability mass function: Binomial distribution for with and as in Pascal's triangle The probability that a ball in a Galton box with 8 layers ( 2/9

3 . ) ends up in the central bin ( ) is for k = 0, 1, 2,..., n, where is the binomial coefficient (hence the name of the distribution) "n choose k", also denoted C(n, k), n C k, or n C k. 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 Looking at the expression ƒ(k, n, p) as a function of k, there is a k value that maximizes it. This k value can be found by calculating and comparing it to 1. There is always an integer M that satisfies ƒ(k, n, p) is monotone increasing for k < M and monotone decreasing for k > M, with the exception of the case where (n + 1)p is an integer. In this case, there are two values for which ƒ is maximal: (n + 1)p and (n + 1)p 1. M is the most probable (most likely) outcome of the Bernoulli trials and is called the mode. Note that the probability of it occurring can be fairly small. Cumulative distribution function The cumulative distribution function can be expressed as: where is the "floor" under x, i.e. the greatest integer less than or equal to x. 3/9

4 It can also be represented in terms of the regularized incomplete beta function, as follows: For k np, upper bounds for the lower tail of the distribution function can be derived. In particular, Hoeffding's inequality yields the bound and Chernoff's inequality can be used to derive the bound Moreover, these bounds are reasonably tight when p = 1/2, since the following expression holds for all k 3n/8 [1] Mean and variance If X ~ B(n, p) (that is, X is a binomially distributed random variable), then the expected value of X is and the variance is Mode and median Usually the mode of a binomial B(n, p) distribution is equal to, where is the floor function. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows: 4/9

5 In general, there is no single formula to find the median for a binomial distribution, and it may even be nonunique. However several special results have been established: If np is an integer, then the mean, median, and mode coincide and equal np. [2][3] Any median m must lie within the interval np m np. [4] A median m cannot lie too far away from the mean: m np min{ ln 2, max{p, 1 p} }. [5] The median is unique and equal to m = round(np) in cases when either p 1 ln 2 or p ln 2 or m np min{p, 1 p} (except for the case when p = ½ and n is odd). [4][5] When p = 1/2 and n is odd, any number m in the interval ½(n 1) m ½(n + 1) is a median of the binomial distribution. If p = 1/2 and n is even, then m = n/2 is the unique median. Covariance between two binomials If two binomially distributed random variables X and Y are observed together, estimating their covariance can be useful. Using the definition of covariance, in the case n = 1 (thus being Bernoulli trials) we have The first term is non zero only when both X and Y are one, and μ X and μ Y are equal to the two probabilities. Defining p B as the probability of both happening at the same time, this gives and for n such trials again due to independence If X and Y are the same variable, this reduces to the variance formula given above. Relationship to other distributions Sums of binomials If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables with the same probability p, then X + Y is again a binomial variable; its distribution is Conditional binomials 5/9

6 If X ~ B(n, p) and, conditional on X, Y ~ B(X, q), then Y is a simple binomial variable with distribution Bernoulli distribution The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B(1, p) has the same meaning as X ~ Bern(p). Conversely, any binomial distribution, B(n, p), is the sum of n independent Bernoulli trials, Bern(p), each with the same probability p. Poisson binomial distribution The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent non identical Bernoulli trials Bern(p i ). If X has the Poisson binomial distribution with p 1 = = p n =p then X ~ B(n, p). Normal approximation If n is large enough, then the skew of the distribution is not too great. In this case a resonable approximation to B(n, p) is given by the normal distribution and this basic approximation can be improved in a simple way by using a suitable continuity correction. The basic approximation generally improves as n increases (at least 20) and is better when p is not near to 0 or 1. [6] Various rules of thumb may be used to decide whether n is large enough, and p is far enough from the extremes of zero or one: One rule is that both x=np and n(1 p) must be greater than 5. However, the specific number varies from source Binomial PDF and normal approximation to source, and depends on how good an approximation for n = 6 and p = 0.5 one wants; some sources give 10 which gives virtually the same results as the following rule for large n until n is very large (ex: x=11, n=7752). A second rule [6] is that for n > 5 the normal approximation is adequate if Another commonly used rule holds that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values, that is if 6/9

7 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; the uncorrected normal approximation gives considerably less accurate results. This approximation, known as de Moivre Laplace theorem, is a huge time saver when undertaking calculations by hand (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 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 Bernoulli variables with parameter p. This fact is the basis of a hypothesis test, a "proportion z test," for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic. [7] For example, suppose one randomly samples 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 groups of n people were sampled 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, as a proportion of the expected value, gets smaller, which allows a more precise estimate of the unknown parameter p. Poisson approximation The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed. Therefore the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to two rules of thumb, this approximation is good if n 20 and p 0.05, or if n 100 and np 10. [8] Limiting distributions Poisson limit theorem: 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. de Moivre Laplace theorem: As n approaches while p remains fixed, the distribution of approaches the normal distribution with expected value 0 and variance 1. This result is sometimes loosely stated by saying that the distribution of X is asymtotically normal with expected 7/9

9 apply. See Terms of use for details. Wikipedia is a registered trademark of the Wikimedia Foundation, Inc., a non profit organization. 9/9

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