{"history":[],"article":{"id":"b29ce31d-9871-47c6-95b9-f9de2573ff3f","title":"Binomial distribution","slug":"binomial-distribution","content":":::summary\n:::\n\n# \n[[File:Pascal's triangle; binomial distribution.svg|thumb|280px|Binomial distribution for\nwith n and k as in [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle)\nThe probability that a ball in a [Galton box](https://bliptext.com/articles/bean-machine) with 8 layers (\nends up in the central bin (\nis\n70/256\n.]]\nIn [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics), the **binomial distribution** with parameters n and p is the [discrete probability distribution](https://bliptext.com/articles/discrete-probability-distribution) of the number of successes in a sequence of n [independent](https://bliptext.com/articles/statistical-independence) [experiments](https://bliptext.com/articles/experiment-probability-theory), each asking a [yes–no question](https://bliptext.com/articles/yes-no-question), and each with its own [Boolean](https://bliptext.com/articles/boolean-valued-function)-valued [outcome](https://bliptext.com/articles/outcome-probability): success (with probability p) or failure (with probability\n). A single success/failure experiment is also called a [Bernoulli trial](https://bliptext.com/articles/bernoulli-trial) or Bernoulli experiment, and a sequence of outcomes is called a [Bernoulli process](https://bliptext.com/articles/bernoulli-process); for a single trial, i.e.,\n, the binomial distribution is a [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution).\nThe binomial distribution is the basis for the [binomial test](https://bliptext.com/articles/binomial-test) of [statistical significance](https://bliptext.com/articles/statistical-significance).\nThe binomial distribution is frequently used to model the number of successes in a sample of size n drawn [with replacement](https://bliptext.com/articles/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](https://bliptext.com/articles/hypergeometric-distribution), not a binomial one.\nHowever, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.\n\n# Definitions\n\n## Probability mass function\nIf the [random variable](https://bliptext.com/articles/random-variable) X follows the binomial distribution with parameters\nn ∈ \\mathbb{N}\nand\np ∈\n, we write\nX ~ B(n, p)\n. The probability of getting exactly k successes in n independent Bernoulli trials (with the same rate p) is given by the [probability mass function](https://bliptext.com/articles/probability-mass-function):\nfor\n, where\nis the [binomial coefficient](https://bliptext.com/articles/binomial-coefficient).\nThe formula can be understood as follows:\npk qn−k\nis the probability of obtaining the sequence of n independent Bernoulli trials in which k trials are \"successes\" and the remaining\nn − k\ntrials result in \"failure\".\nSince the trials are independent with probabilities remaining constant between them, any sequence of n trials with k successes (and\nn − k\nfailures) has the same probability of being achieved (regardless of positions of successes within the sequence). There are such sequences, since the binomial coefficient counts the number of ways to choose the positions of the k successes among the n trials. The binomial distribution is concerned with the probability of obtaining any of these sequences, meaning the probability of obtaining one of them (\npk qn−k\n) must be added times, hence.\nIn creating reference tables for binomial distribution probability, usually, the table is filled in up to\nn/2\nvalues.\nThis is because for\nk > n/2\n, the probability can be calculated by its complement as\nLooking at the expression\nf(k, n, p)\nas a function of k, there is a k value that maximizes it.\nThis k value can be found by calculating\nand comparing it to 1.\nThere is always an integer M that satisfies\nf(k, n, p)\nis monotone increasing for\nk < M\nand monotone decreasing for\nk > M\n, with the exception of the case where\n(n + 1)p\nis an integer.\nIn this case, there are two values for which f is maximal:\n(n + 1) p\nand\n(n + 1) p − 1\n. M is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the [mode](https://bliptext.com/articles/mode-statistics).\nEquivalently,\nM − p < np ≤ M + 1 − p\n. Taking the [floor function](https://bliptext.com/articles/floor-and-ceiling-functions), we obtain\n.\n\n## Example\nSuppose a [biased coin](https://bliptext.com/articles/fair-coin) comes up heads with probability 0.3 when tossed.\nThe probability of seeing exactly 4 heads in 6 tosses is\n\n## Cumulative distribution function\nThe [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) can be expressed as:\nwhere is the \"floor\" under\nk\n, i.e. the [greatest integer](https://bliptext.com/articles/floor-and-ceiling-functions) less than or equal to\nk\n.\nIt can also be represented in terms of the [regularized incomplete beta function](https://bliptext.com/articles/regularized-incomplete-beta-function), as follows:\nwhich is equivalent to the [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) of the [F-distribution](https://bliptext.com/articles/f-distribution):\nSome closed-form bounds for the cumulative distribution function are given [below](https://bliptext.com/articles/).\n\n# Properties\n\n## Expected value and variance\nIf\nX ~ B(n, p)\n, that is,\nX\nis a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding a successful result, then the [expected value](https://bliptext.com/articles/expected-value) of\nX\nis:\nThis follows from the linearity of the expected value along with the fact that X is the sum of n identical Bernoulli random variables, each with expected value p. In other words, if are identical (and independent) Bernoulli random variables with parameter p, then\nX = X1 + ...\n+ Xn\nand\nThe [variance](https://bliptext.com/articles/variance) is:\nThis similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances.\n\n## Higher moments\nThe first 6 [central moments](https://bliptext.com/articles/central-moment), defined as, are given by\nThe non-central moments satisfy\nand in general\nwhere are the [Stirling numbers of the second kind](https://bliptext.com/articles/stirling-numbers-of-the-second-kind), and is the kth [falling power](https://bliptext.com/articles/falling-and-rising-factorials) of n. A simple bound\nfollows by bounding the Binomial moments via the [higher Poisson moments](https://bliptext.com/articles/poisson-distribution):\nThis shows that if, then is at most a constant factor away from\n\n## Mode\nUsually the [mode](https://bliptext.com/articles/mode-statistics) of a binomial\nB(n, p)\ndistribution is equal to, where is the [floor function](https://bliptext.com/articles/floor-function).\nHowever, when\n(n + 1)p\nis an integer and\np\nis neither 0 nor 1, then the distribution has two modes:\n(n + 1)p\nand\n(n + 1)p − 1\n. When\np\nis equal to 0 or 1, the mode will be 0 and\nn\ncorrespondingly.\nThese cases can be summarized as follows:\n**Proof:** Let\nFor p=0 only f(0) has a nonzero value with f(0)=1.\nFor p=1 we find f(n)=1 and f(k)=0 for k\\neq n. This proves that the mode is 0 for p=0 and n for p=1.\nLet 0 < p < 1.\nWe find\nFrom this follows\nSo when (n+1)p-1 is an integer, then (n+1)p-1 and (n+1)p is a mode.\nIn the case that, then only is a mode.\n\n## Median\nIn general, there is no single formula to find the [median](https://bliptext.com/articles/median) for a binomial distribution, and it may even be non-unique.\nHowever, several special results have been established:\nnp\nis an integer, then the mean, median, and mode coincide and equal\nnp\n.\nm\nmust lie within the interval.\nm\ncannot lie too far away from the mean:.\nm = round(np)\nwhen\n≤ min\n(except for the case when\np = 1/2\nand\nn\nis odd).\np\nis a rational number (with the exception of\np = 1/2\n\\ and\nn\nodd) the median is unique.\nn\nis odd, any number\nm\nin the interval is a median of the binomial distribution.\nIf and\nn\nis even, then is the unique median.\n\n## Tail bounds\nFor\nk ≤ np\n, upper bounds can be derived for the lower tail of the cumulative distribution function, the probability that there are at most\nk\nsuccesses.\nSince, these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for\nk ≥ np\n.\n[Hoeffding's inequality](https://bliptext.com/articles/hoeffding-s-inequality) yields the simple bound\nwhich is however not very tight.\nIn particular, for\np = 1\n, we have that\nF(k; n, p) = 0\n(for fixed\nk\n,\nn\nwith\nk < n\n), but Hoeffding's bound evaluates to a positive constant.\nA sharper bound can be obtained from the [Chernoff bound](https://bliptext.com/articles/chernoff-bound):\nwhere\nD(a ∥ p)\nis the [relative entropy (or Kullback-Leibler divergence)](https://bliptext.com/articles/kullback-leibler-divergence) between an\na\n-coin and a\np\n-coin (i.e. between the\nBernoulli(a)\nand\nBernoulli(p)\ndistribution):\nAsymptotically, this bound is reasonably tight; see for details.\nOne can also obtain lower bounds on the tail\nF(k; n, p)\n, known as anti-concentration bounds.\nBy approximating the binomial coefficient with [Stirling's formula](https://bliptext.com/articles/stirling-s-approximation) it can be shown that\nwhich implies the simpler but looser bound\nFor\np = 1/2\nand\nk ≥ 3n/8\nfor even\nn\n, it is possible to make the denominator constant:\n\n# Statistical inference\n\n## Estimation of parameters\nWhen\nn\nis known, the parameter\np\ncan be estimated using the proportion of successes:\nThis estimator is found using [maximum likelihood estimator](https://bliptext.com/articles/maximum-likelihood-estimator) and also the [method of moments](https://bliptext.com/articles/method-of-moments-statistics).\nThis estimator is [unbiased](https://bliptext.com/articles/bias-of-an-estimator) and uniformly with [minimum variance](https://bliptext.com/articles/minimum-variance-unbiased-estimator), proven using [Lehmann–Scheffé theorem](https://bliptext.com/articles/lehmann-scheff-theorem), since it is based on a [minimal sufficient](https://bliptext.com/articles/minimal-sufficient) and [complete](https://bliptext.com/articles/completeness-statistics) statistic (i.e.:\nx\n). It is also [consistent](https://bliptext.com/articles/consistent-estimator) both in probability and in [MSE](https://bliptext.com/articles/[mean](https://bliptext.com/articles/arithmetic-mean)-squared-error). This statistic is [asymptotically](https://bliptext.com/articles/asymptotic-distribution) [normal](https://bliptext.com/articles/normal-distribution) thanks to the [central limit theorem](https://bliptext.com/articles/central-limit-theorem), because it is the same as taking the [mean](https://bliptext.com/articles/arithmetic-mean) over Bernoulli samples. It has a variance of, a property which is used in various ways, such as in [Wald's confidence intervals](https://bliptext.com/articles/binomial-proportion-confidence-interval).\nA closed form [Bayes estimator](https://bliptext.com/articles/bayes-estimator) for\np\nalso exists when using the [Beta distribution](https://bliptext.com/articles/beta-distribution) as a [conjugate](https://bliptext.com/articles/conjugate-prior) [prior distribution](https://bliptext.com/articles/prior-distribution).\nWhen using a general as a prior, the [posterior mean](https://bliptext.com/articles/bayes-estimator) estimator is:\nThe Bayes estimator is [asymptotically efficient](https://bliptext.com/articles/asymptotic-efficiency-bayes) and as the sample size approaches infinity (\nn → ∞\n), it approaches the [MLE](https://bliptext.com/articles/maximum-likelihood-estimation) solution. The Bayes estimator is [biased](https://bliptext.com/articles/bias-of-an-estimator) (how much depends on the priors), [admissible](https://bliptext.com/articles/bayes-estimator) and [consistent](https://bliptext.com/articles/consistent-estimator) in probability. Using the Bayesian estimator with the Beta distribution can be used with [Thompson sampling](https://bliptext.com/articles/thompson-sampling).\nFor the special case of using the [standard uniform distribution](https://bliptext.com/articles/standard-uniform-distribution) as a [non-informative prior](https://bliptext.com/articles/non-informative-prior),, the posterior mean estimator becomes:\n(A [posterior mode](https://bliptext.com/articles/bayes-estimator) should just lead to the standard estimator.) This method is called the [rule of succession](https://bliptext.com/articles/rule-of-succession), which was introduced in the 18th century by [Pierre-Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace).\nWhen relying on [Jeffreys prior](https://bliptext.com/articles/jeffreys-prior), the prior is, which leads to the estimator:\nWhen estimating\np\nwith very rare events and a small\nn\n(e.g.: if\nx = 0\n), then using the standard estimator leads to which sometimes is unrealistic and undesirable. In such cases there are various alternative estimators. One way is to use the Bayes estimator, leading to: Another method is to use the upper bound of the [confidence interval](https://bliptext.com/articles/confidence-interval) obtained using the [rule of three](https://bliptext.com/articles/rule-of-three-statistics):\n\n## Confidence intervals for the parameter p\nEven for quite large values of n, the actual distribution of the mean is significantly nonnormal.\nBecause of this problem several methods to estimate confidence intervals have been proposed.\nIn the equations for confidence intervals below, the variables have the following meaning:\n\n### Wald method\nA [continuity correction](https://bliptext.com/articles/continuity-correction) of\n0.5/n\nmay be added.\n\n### Agresti–Coull method\nHere the estimate of\np\nis modified to\nThis method works well for\nn > 10\nand\nn1 ≠ 0, n\n. See here for n\\leq 10.\nFor\nn1 = 0, n\nuse the Wilson (score) method below.\n\n### Arcsine method\n\n### Wilson (score) method\nThe notation in the formula below differs from the previous formulas in two respects:\nzx\nhas a slightly different interpretation in the formula below: it has its ordinary meaning of 'the\nx\nth quantile of the standard normal distribution', rather than being a shorthand for 'the\n(1 − x)\nth quantile'.\n\n### Comparison\nThe so-called \"exact\" ([Clopper–Pearson](https://bliptext.com/articles/binomial-proportion-confidence-interval)) method is the most conservative.\n(Exact does not mean perfectly accurate; rather, it indicates that the estimates will not be less conservative than the true value.)\nThe Wald method, although commonly recommended in textbooks, is the most biased.\n\n# Related distributions\n\n## Sums of binomials\nIf\nX ~ B(n, p)\nand\nY ~ B(m, p)\nare independent binomial variables with the same probability\np\n, then\nX + Y\nis again a binomial variable; its distribution is\nZ = X + Y ~ B(n + m, p)\nA Binomial distributed random variable\nX ~ B(n, p)\ncan be considered as the sum of\nn\nBernoulli distributed random variables.\nSo the sum of two Binomial distributed random variables\nX ~ B(n, p)\nand\nY ~ B(m, p)\nis equivalent to the sum of\nn + m\nBernoulli distributed random variables, which means\nZ = X + Y ~ B(n + m, p)\n. This can also be proven directly using the addition rule.\nHowever, if\nX\nand\nY\ndo not have the same probability\np\n, then the variance of the sum will be [smaller than the variance of a binomial variable](https://bliptext.com/articles/binomial-sum-variance-inequality) distributed as\nB(n + m, p)\n.\n\n## Poisson binomial distribution\nThe binomial distribution is a special case of the [Poisson binomial distribution](https://bliptext.com/articles/poisson-binomial-distribution), which is the distribution of a sum of\nn\nindependent non-identical [Bernoulli trials](https://bliptext.com/articles/bernoulli-trials)\nB(pi)\n.\n\n## Ratio of two binomial distributions\nThis result was first derived by Katz and coauthors in 1978.\nLet X ~ B(n, p1) and Y ~ B(m, p2) be independent.\nLet T = (X/n) / (Y/m).\nThen log(T) is approximately normally distributed with mean log(p1/p2) and variance ((1/p1) − 1)/n + ((1/p2) − 1)/m.\n\n## Conditional binomials\nIf X ~ B(n, p) and Y | X ~ B(X, q) (the conditional distribution of Y, given X), then Y is a simple binomial random variable with distribution Y ~ B(n, pq).\nFor example, imagine throwing n balls to a basket UX and taking the balls that hit and throwing them to another basket UY.\nIf p is the probability to hit UX then X ~ B(n, p) is the number of balls that hit UX.\nIf q is the probability to hit UY then the number of balls that hit UY is Y ~ B(X, q) and therefore Y ~ B(n, pq).\nSince and, by the [law of total probability](https://bliptext.com/articles/law-of-total-probability),\nSince the equation above can be expressed as Factoring and pulling all the terms that don't depend on k out of the sum now yields\nAfter substituting i = k - m in the expression above, we get\nNotice that the sum (in the parentheses) above equals by the [binomial theorem](https://bliptext.com/articles/binomial-theorem).\nSubstituting this in finally yields\nand thus as desired.\n\n## Bernoulli distribution\nThe [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) is a special case of the binomial distribution, where\nn = 1\n. Symbolically,\nX ~ B(1, p)\nhas the same meaning as\nX ~ Bernoulli(p)\n. Conversely, any binomial distribution,\nB(n, p)\n, is the distribution of the sum of\nn\nindependent [Bernoulli trials](https://bliptext.com/articles/bernoulli-trials),\nBernoulli(p)\n, each with the same probability\np\n.\n\n## Normal approximation\nBinomial Distribution.svg and normal [probability density function](https://bliptext.com/articles/probability-density-function) approximation for\nn = 6\nand\np = 0.5\n]]\nIf\nn\nis large enough, then the skew of the distribution is not too great.\nIn this case a reasonable approximation to\nB(n, p)\nis given by the [normal distribution](https://bliptext.com/articles/normal-distribution)\nand this basic approximation can be improved in a simple way by using a suitable [continuity correction](https://bliptext.com/articles/continuity-correction).\nThe basic approximation generally improves as\nn\nincreases (at least 20) and is better when\np\nis not near to 0 or 1.\nVarious [rules of thumb](https://bliptext.com/articles/rule-of-thumb) may be used to decide whether\nn\nis large enough, and\np\nis far enough from the extremes of zero or one:\nn > 5\nthe normal approximation is adequate if the absolute value of the skewness is strictly less than 0.3; that is, if This can be made precise using the [Berry–Esseen theorem](https://bliptext.com/articles/berry-esseen-theorem).\nThe rule is totally equivalent to request that\nMoving terms around yields:\nSince 09.\nOn the other hand, apply again the square root and divide by 3,\nSubtracting the second set of inequalities from the first one yields:\nand so, the desired first rule is satisfied,\nnp\nand\nn(1 − p)\nmust be greater than or equal to 5.\nHowever, the specific number varies from source to source, and depends on how good an approximation one wants.\nIn particular, if one uses 9 instead of 5, the rule implies the results stated in the previous paragraphs.\nAssume that both values np and n(1-p) are greater than 9.\nSince 0< p<1, we easily have that\nWe only have to divide now by the respective factors p and 1-p, to deduce the alternative form of the 3-standard-deviation rule:\nThe following is an example of applying a [continuity correction](https://bliptext.com/articles/continuity-correction).\nSuppose one wishes to calculate\nPr(X ≤ 8)\nfor a binomial random variable\nX\n. If\nY\nhas a distribution given by the normal approximation, then\nPr(X ≤ 8)\nis approximated by\nPr(Y ≤ 8.5)\n. The addition of 0.5 is the continuity correction; the uncorrected normal approximation gives considerably less accurate results.\nThis approximation, known as [de Moivre–Laplace theorem](https://bliptext.com/articles/de-moivre-laplace-theorem), is a huge time-saver when undertaking calculations by hand (exact calculations with large\nn\nare very onerous); historically, it was the first use of the normal distribution, introduced in [Abraham de Moivre](https://bliptext.com/articles/abraham-de-moivre)'s book [The Doctrine of Chances](https://bliptext.com/articles/the-doctrine-of-chances) in 1738. Nowadays, it can be seen as a consequence of the [central limit theorem](https://bliptext.com/articles/central-limit-theorem) since\nB(n, p)\nis a sum of\nn\nindependent, identically distributed [Bernoulli variables](https://bliptext.com/articles/bernoulli-distribution) with parameter\np\n. This fact is the basis of a [hypothesis test](https://bliptext.com/articles/hypothesis-test), a \"proportion z-test\", for the value of\np\nusing\nx/n\n, the sample proportion and estimator of\np\n, in a [common test statistic](https://bliptext.com/articles/common-test-statistics).\nFor example, suppose one randomly samples\nn\npeople out of a large population and ask them whether they agree with a certain statement.\nThe proportion of people who agree will of course depend on the sample.\nIf 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\n\n## Poisson approximation\nThe binomial distribution converges towards the [Poisson distribution](https://bliptext.com/articles/poisson-distribution) as the number of trials goes to infinity while the product\nnp\nconverges to a finite limit.\nTherefore, the Poisson distribution with parameter\nλ = np\ncan be used as an approximation to\nB(n, p)\nof the binomial distribution if\nn\nis sufficiently large and\np\nis sufficiently small.\nAccording to rules of thumb, this approximation is good if\nn ≥ 20\nand\np ≤ 0.05\nsuch that\nnp ≤ 1\n, or if\nn > 50\nand\np < 0.1\nsuch that\nnp < 5\n, or if\nn ≥ 100\nand\nnp ≤ 10\n.\nConcerning the accuracy of Poisson approximation, see Novak, ch.\n4, and references therein.\n\n## Limiting distributions\nn\napproaches\n∞\nand\np\napproaches 0 with the product\nnp\nheld fixed, the\nBinomial(n, p)\ndistribution approaches the [Poisson distribution](https://bliptext.com/articles/poisson-distribution) with [expected value](https://bliptext.com/articles/expected-value)\nλ = np\n.\nn\napproaches\n∞\nwhile\np\nremains fixed, the distribution of\nX\nis [asymptotically normal](https://bliptext.com/articles/asymptotic-normality) with expected value 0 and [variance](https://bliptext.com/articles/variance) 1.\nThis result is a specific case of the [central limit theorem](https://bliptext.com/articles/central-limit-theorem).\n\n## Beta distribution\nThe binomial distribution and beta distribution are different views of the same model of repeated Bernoulli trials.\nThe binomial distribution is the [PMF](https://bliptext.com/articles/probability-mass-function) of k successes given n independent events each with a probability p of success.\nMathematically, when\nα = k + 1\nand\nβ = n − k + 1\n, the beta distribution and the binomial distribution are related by a factor of\nn + 1\n[Beta distributions](https://bliptext.com/articles/beta-distribution) also provide a family of [prior probability distributions](https://bliptext.com/articles/prior-distribution) for binomial distributions in [Bayesian inference](https://bliptext.com/articles/bayesian-inference):\nGiven a uniform prior, the posterior distribution for the probability of success p given n independent events with k observed successes is a beta distribution.\n\n# Computational methods\n\n## Random number generation\nMethods for [random number generation](https://bliptext.com/articles/random-number-generation) where the [marginal distribution](https://bliptext.com/articles/marginal-distribution) is a binomial distribution are well-established.\nOne way to generate [random variates](https://bliptext.com/articles/random-variate) samples from a binomial distribution is to use an inversion algorithm.\nTo do so, one must calculate the probability that\nPr(X = k)\nfor all values k from\n0\nthrough n. (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a [pseudorandom number generator](https://bliptext.com/articles/pseudorandom-number-generator) to generate samples uniformly between 0 and 1, one can transform the calculated samples into discrete numbers by using the probabilities calculated in the first step.\n\n# History\nThis distribution was derived by [Jacob Bernoulli](https://bliptext.com/articles/jacob-bernoulli).\nHe considered the case where\np = r/(r + s)\nwhere\np\nis the probability of success and\nr\nand\ns\nare positive integers.\n[Blaise Pascal](https://bliptext.com/articles/blaise-pascal) had earlier considered the case where\np = 1/2\n, tabulating the corresponding binomial coefficients in what is now recognized as [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:30:59.556Z","updated_at":"2025-01-25T16:30:59.556Z","search_vector":"'binomi':1 'distribut':2","search_content":" \n \n\n# \n[[File:Pascal's triangle; binomial distribution.svg|thumb|280px|Binomial distribution for\nwith n and k as in [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle)\nThe probability that a ball in a [Galton box](https://bliptext.com/articles/bean-machine) with 8 layers (\nends up in the central bin (\nis\n70/256\n.]]\nIn [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics), the **binomial distribution** with parameters n and p is the [discrete probability distribution](https://bliptext.com/articles/discrete-probability-distribution) of the number of successes in a sequence of n [independent](https://bliptext.com/articles/statistical-independence) [experiments](https://bliptext.com/articles/experiment-probability-theory), each asking a [yes–no question](https://bliptext.com/articles/yes-no-question), and each with its own [Boolean](https://bliptext.com/articles/boolean-valued-function)-valued [outcome](https://bliptext.com/articles/outcome-probability): success (with probability p) or failure (with probability\n). A single success/failure experiment is also called a [Bernoulli trial](https://bliptext.com/articles/bernoulli-trial) or Bernoulli experiment, and a sequence of outcomes is called a [Bernoulli process](https://bliptext.com/articles/bernoulli-process); for a single trial, i.e.,\n, the binomial distribution is a [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution).\nThe binomial distribution is the basis for the [binomial test](https://bliptext.com/articles/binomial-test) of [statistical significance](https://bliptext.com/articles/statistical-significance).\nThe binomial distribution is frequently used to model the number of successes in a sample of size n drawn [with replacement](https://bliptext.com/articles/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](https://bliptext.com/articles/hypergeometric-distribution), not a binomial one.\nHowever, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.\n\n# Definitions\n\n## Probability mass function\nIf the [random variable](https://bliptext.com/articles/random-variable) X follows the binomial distribution with parameters\nn ∈ \\mathbb{N}\nand\np ∈\n, we write\nX ~ B(n, p)\n. The probability of getting exactly k successes in n independent Bernoulli trials (with the same rate p) is given by the [probability mass function](https://bliptext.com/articles/probability-mass-function):\nfor\n, where\nis the [binomial coefficient](https://bliptext.com/articles/binomial-coefficient).\nThe formula can be understood as follows:\npk qn−k\nis the probability of obtaining the sequence of n independent Bernoulli trials in which k trials are \"successes\" and the remaining\nn − k\ntrials result in \"failure\".\nSince the trials are independent with probabilities remaining constant between them, any sequence of n trials with k successes (and\nn − k\nfailures) has the same probability of being achieved (regardless of positions of successes within the sequence). There are such sequences, since the binomial coefficient counts the number of ways to choose the positions of the k successes among the n trials. The binomial distribution is concerned with the probability of obtaining any of these sequences, meaning the probability of obtaining one of them (\npk qn−k\n) must be added times, hence.\nIn creating reference tables for binomial distribution probability, usually, the table is filled in up to\nn/2\nvalues.\nThis is because for\nk > n/2\n, the probability can be calculated by its complement as\nLooking at the expression\nf(k, n, p)\nas a function of k, there is a k value that maximizes it.\nThis k value can be found by calculating\nand comparing it to 1.\nThere is always an integer M that satisfies\nf(k, n, p)\nis monotone increasing for\nk M\n, with the exception of the case where\n(n + 1)p\nis an integer.\nIn this case, there are two values for which f is maximal:\n(n + 1) p\nand\n(n + 1) p − 1\n. M is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the [mode](https://bliptext.com/articles/mode-statistics).\nEquivalently,\nM − p 10\nand\nn1 ≠ 0, n\n. See here for n\\leq 10.\nFor\nn1 = 0, n\nuse the Wilson (score) method below.\n\n### Arcsine method\n\n### Wilson (score) method\nThe notation in the formula below differs from the previous formulas in two respects:\nzx\nhas a slightly different interpretation in the formula below: it has its ordinary meaning of 'the\nx\nth quantile of the standard normal distribution', rather than being a shorthand for 'the\n(1 − x)\nth quantile'.\n\n### Comparison\nThe so-called \"exact\" ([Clopper–Pearson](https://bliptext.com/articles/binomial-proportion-confidence-interval)) method is the most conservative.\n(Exact does not mean perfectly accurate; rather, it indicates that the estimates will not be less conservative than the true value.)\nThe Wald method, although commonly recommended in textbooks, is the most biased.\n\n# Related distributions\n\n## Sums of binomials\nIf\nX ~ B(n, p)\nand\nY ~ B(m, p)\nare independent binomial variables with the same probability\np\n, then\nX + Y\nis again a binomial variable; its distribution is\nZ = X + Y ~ B(n + m, p)\nA Binomial distributed random variable\nX ~ B(n, p)\ncan be considered as the sum of\nn\nBernoulli distributed random variables.\nSo the sum of two Binomial distributed random variables\nX ~ B(n, p)\nand\nY ~ B(m, p)\nis equivalent to the sum of\nn + m\nBernoulli distributed random variables, which means\nZ = X + Y ~ B(n + m, p)\n. This can also be proven directly using the addition rule.\nHowever, if\nX\nand\nY\ndo not have the same probability\np\n, then the variance of the sum will be [smaller than the variance of a binomial variable](https://bliptext.com/articles/binomial-sum-variance-inequality) distributed as\nB(n + m, p)\n.\n\n## Poisson binomial distribution\nThe binomial distribution is a special case of the [Poisson binomial distribution](https://bliptext.com/articles/poisson-binomial-distribution), which is the distribution of a sum of\nn\nindependent non-identical [Bernoulli trials](https://bliptext.com/articles/bernoulli-trials)\nB(pi)\n.\n\n## Ratio of two binomial distributions\nThis result was first derived by Katz and coauthors in 1978.\nLet X ~ B(n, p1) and Y ~ B(m, p2) be independent.\nLet T = (X/n) / (Y/m).\nThen log(T) is approximately normally distributed with mean log(p1/p2) and variance ((1/p1) − 1)/n + ((1/p2) − 1)/m.\n\n## Conditional binomials\nIf X ~ B(n, p) and Y | X ~ B(X, q) (the conditional distribution of Y, given X), then Y is a simple binomial random variable with distribution Y ~ B(n, pq).\nFor example, imagine throwing n balls to a basket UX and taking the balls that hit and throwing them to another basket UY.\nIf p is the probability to hit UX then X ~ B(n, p) is the number of balls that hit UX.\nIf q is the probability to hit UY then the number of balls that hit UY is Y ~ B(X, q) and therefore Y ~ B(n, pq).\nSince and, by the [law of total probability](https://bliptext.com/articles/law-of-total-probability),\nSince the equation above can be expressed as Factoring and pulling all the terms that don't depend on k out of the sum now yields\nAfter substituting i = k - m in the expression above, we get\nNotice that the sum (in the parentheses) above equals by the [binomial theorem](https://bliptext.com/articles/binomial-theorem).\nSubstituting this in finally yields\nand thus as desired.\n\n## Bernoulli distribution\nThe [Bernoulli distribution](https://bliptext.com/articles/bernoulli-distribution) is a special case of the binomial distribution, where\nn = 1\n. Symbolically,\nX ~ B(1, p)\nhas the same meaning as\nX ~ Bernoulli(p)\n. Conversely, any binomial distribution,\nB(n, p)\n, is the distribution of the sum of\nn\nindependent [Bernoulli trials](https://bliptext.com/articles/bernoulli-trials),\nBernoulli(p)\n, each with the same probability\np\n.\n\n## Normal approximation\nBinomial Distribution.svg and normal [probability density function](https://bliptext.com/articles/probability-density-function) approximation for\nn = 6\nand\np = 0.5\n]]\nIf\nn\nis large enough, then the skew of the distribution is not too great.\nIn this case a reasonable approximation to\nB(n, p)\nis given by the [normal distribution](https://bliptext.com/articles/normal-distribution)\nand this basic approximation can be improved in a simple way by using a suitable [continuity correction](https://bliptext.com/articles/continuity-correction).\nThe basic approximation generally improves as\nn\nincreases (at least 20) and is better when\np\nis not near to 0 or 1.\nVarious [rules of thumb](https://bliptext.com/articles/rule-of-thumb) may be used to decide whether\nn\nis large enough, and\np\nis far enough from the extremes of zero or one:\nn > 5\nthe normal approximation is adequate if the absolute value of the skewness is strictly less than 0.3; that is, if This can be made precise using the [Berry–Esseen theorem](https://bliptext.com/articles/berry-esseen-theorem).\nThe rule is totally equivalent to request that\nMoving terms around yields:\nSince 0 9.\nOn the other hand, apply again the square root and divide by 3,\nSubtracting the second set of inequalities from the first one yields:\nand so, the desired first rule is satisfied,\nnp\nand\nn(1 − p)\nmust be greater than or equal to 5.\nHowever, the specific number varies from source to source, and depends on how good an approximation one wants.\nIn particular, if one uses 9 instead of 5, the rule implies the results stated in the previous paragraphs.\nAssume that both values np and n(1-p) are greater than 9.\nSince 0 50\nand\np < 0.1\nsuch that\nnp < 5\n, or if\nn ≥ 100\nand\nnp ≤ 10\n.\nConcerning the accuracy of Poisson approximation, see Novak, ch.\n4, and references therein.\n\n## Limiting distributions\nn\napproaches\n∞\nand\np\napproaches 0 with the product\nnp\nheld fixed, the\nBinomial(n, p)\ndistribution approaches the [Poisson distribution](https://bliptext.com/articles/poisson-distribution) with [expected value](https://bliptext.com/articles/expected-value)\nλ = np\n.\nn\napproaches\n∞\nwhile\np\nremains fixed, the distribution of\nX\nis [asymptotically normal](https://bliptext.com/articles/asymptotic-normality) with expected value 0 and [variance](https://bliptext.com/articles/variance) 1.\nThis result is a specific case of the [central limit theorem](https://bliptext.com/articles/central-limit-theorem).\n\n## Beta distribution\nThe binomial distribution and beta distribution are different views of the same model of repeated Bernoulli trials.\nThe binomial distribution is the [PMF](https://bliptext.com/articles/probability-mass-function) of k successes given n independent events each with a probability p of success.\nMathematically, when\nα = k + 1\nand\nβ = n − k + 1\n, the beta distribution and the binomial distribution are related by a factor of\nn + 1\n[Beta distributions](https://bliptext.com/articles/beta-distribution) also provide a family of [prior probability distributions](https://bliptext.com/articles/prior-distribution) for binomial distributions in [Bayesian inference](https://bliptext.com/articles/bayesian-inference):\nGiven a uniform prior, the posterior distribution for the probability of success p given n independent events with k observed successes is a beta distribution.\n\n# Computational methods\n\n## Random number generation\nMethods for [random number generation](https://bliptext.com/articles/random-number-generation) where the [marginal distribution](https://bliptext.com/articles/marginal-distribution) is a binomial distribution are well-established.\nOne way to generate [random variates](https://bliptext.com/articles/random-variate) samples from a binomial distribution is to use an inversion algorithm.\nTo do so, one must calculate the probability that\nPr(X = k)\nfor all values k from\n0\nthrough n. (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a [pseudorandom number generator](https://bliptext.com/articles/pseudorandom-number-generator) to generate samples uniformly between 0 and 1, one can transform the calculated samples into discrete numbers by using the probabilities calculated in the first step.\n\n# History\nThis distribution was derived by [Jacob Bernoulli](https://bliptext.com/articles/jacob-bernoulli).\nHe considered the case where\np = r/(r + s)\nwhere\np\nis the probability of success and\nr\nand\ns\nare positive integers.\n[Blaise Pascal](https://bliptext.com/articles/blaise-pascal) had earlier considered the case where\np = 1/2\n, tabulating the corresponding binomial coefficients in what is now recognized as [Pascal's triangle](https://bliptext.com/articles/pascal-s-triangle).","revisionCount":0},"hasMore":false,"nextCursor":null}