{{Probability distribution

| name = Binomial distribution

| type = mass

| pdf_image = 300px|Probability mass function for the binomial distribution

| cdf_image = 300px|Cumulative distribution function for the binomial distribution

| notation =

| parameters = – number of trials – success probability for each trial

| support = – number of successes

| pdf =

| cdf = (the regularized incomplete beta function)

| mean =

| median = or

| mode = or

| variance =

| skewness =

| kurtosis =

| entropy = in shannons. For nats, use the natural log in the log.

| mgf =

| char =

| pgf =

| fisher = (for fixed )

}}

thumb|280px|Binomial distribution for with and as in [[Pascal's triangleThe probability that a ball in a Galton box with 8 layers () ends up in the central bin () is .]]

In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability ) or failure (with probability ). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process. For a single trial, that is, when , the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the binomial test of statistical significance.

The binomial distribution is frequently used to model the number of successes in a sample of size drawn with replacement from a population of size . 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 much larger than , the binomial distribution remains a good approximation, and is widely used.

Definitions

Probability mass function

If the random variable follows the binomial distribution with parameters (a natural number) and , we write . The probability of getting exactly successes in independent Bernoulli trials (with the same rate ) is given by the probability mass function:

for , where

is the binomial coefficient. The formula can be understood as follows: is the probability of obtaining the sequence of independent Bernoulli trials in which trials are "successes" and the remaining trials are "failures". Since the trials are independent with probabilities remaining constant between them, any sequence of trials with successes (and failures) 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 successes among the trials. The binomial distribution is concerned with the probability of obtaining any of these sequences, meaning the probability of obtaining one of them () must be added times, hence .

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

Looking at the expression as a function of , there is a value that maximizes it. This value can be found by calculating

and comparing it to 1. There is always an integer that satisfies

is monotone increasing for and monotone decreasing for , with the exception of the case where is an integer. In this case, there are two values for which is maximal: and . 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.

Example

Suppose a biased coin comes up heads with probability 0.3 when tossed. The probability of seeing exactly 4 heads in 6 tosses is

Cumulative distribution function

The cumulative distribution function can be expressed as:

where is the "floor" under ; that is, the greatest integer less than or equal to .

It can also be represented in terms of the regularized incomplete beta function, as follows:

<math display="block">\begin{align}

F(k;n,p) & = \Pr(X \le k) \\

&= I_{1-p}(n-k, k+1) \\

& = (n-k) {n \choose k} \int_0^{1-p} t^{n-k-1} (1-t)^k \, dt ,

\end{align}</math>

which is equivalent to the cumulative distribution functions of the beta distribution and of the -distribution:

Some closed-form bounds for the cumulative distribution function are given below.

Properties

Expected value and variance

If , that is, is a binomially distributed random variable, being the total number of experiments and the probability of each experiment yielding a successful result, then the expected value of is:See Proof Wiki

This follows from the linearity of the expected value along with the fact that is the sum of identical Bernoulli random variables, each with expected value . In other words, if are identical (and independent) Bernoulli random variables with parameter , then and

The variance is:

This similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances.

Higher moments

The first 6 central moments, defined as , are given by

<math display="block">\begin{align}

\mu_1 &= 0, \\

\mu_2 &= np \left(1-p\right), \\

\mu_3 &= np \left(1-p\right) \left(1-2p\right), \\

\mu_4 &= np \left(1-p\right) \left[1 + \left(3n-6\right) p \left(1-p\right)\right],\\

\mu_5 &= np \left(1-p\right) \left(1-2p\right) \left[1 + \left(10n-12\right) p \left(1-p\right)\right],\\

\mu_6 &= np \left(1-p\right) \left[1 - 30p\left(1-p\right)\left[1-4p(1-p)\right] + 5np \left(1-p\right)\left[5 - 26p\left(1-p\right)\right] + 15n^2 p^2 \left(1-p\right)^2\right].

\end{align}</math>

The non-central moments satisfy

<math display="block">\begin{align}

\operatorname {E}[X] &= np, \\

\operatorname {E}[X^2] &= np(1-p)+n^2p^2,

\end{align}</math>

and in general<ref name="Andreas2008">

{{citation

|last1=Knoblauch |first1=Andreas

|title=Closed-Form Expressions for the Moments of the Binomial Probability Distribution

|year=2008

|journal=SIAM Journal on Applied Mathematics

|volume=69

|issue=1

|pages=197–204

|doi=10.1137/070700024

|jstor=40233780

}}</ref><ref name="Nguyen2021">

{{citation

|last1=Nguyen |first1=Duy

|title=A probabilistic approach to the moments of binomial random variables and application

|year=2021

|journal=The American Statistician

|url=https://www.tandfonline.com/doi/abs/10.1080/00031305.2019.1679257?journalCode=utas20

|volume=75

|issue=1

|pages=101–103

|doi=10.1080/00031305.2019.1679257

|s2cid=209923008

|url-access=subscription

}}</ref>

<math display="block">

\operatorname {E}[X^c] = \sum_{k=0}^c \left\{ {c \atop k} \right\} n^{\underline{k}} p^k,

</math>

where are the Stirling numbers of the second kind, and is the -th falling power of .

A simple bound

<ref>{{Citation |last1=D. Ahle |first1=Thomas |title=Sharp and Simple Bounds for the raw Moments of the Binomial and Poisson Distributions

|year=2022

|volume=182

|doi=10.1016/j.spl.2021.109306

|journal=Statistics & Probability Letters

|article-number=109306 |arxiv=2103.17027 }}</ref> follows by bounding the Binomial moments via the higher Poisson moments:

<math display="block">

\operatorname {E}[X^c] \le

\left[\frac{c}{\ln\left(1+\frac{c}{np}\right)}\right]^c \le (np)^c \exp\left(\frac{c^2}{2np}\right).

</math>

This shows that if , then is at most a constant factor away from .

The moment-generating function is .

Mode

Usually the mode of a binomial distribution is equal to , where is the floor function. However, when is an integer and is neither 0 nor 1, then the distribution has two modes: and . When is equal to 0 or 1, the mode will be 0 and correspondingly. These cases can be summarized as follows:

<math display="block">\text{mode} =

\begin{cases}

\lfloor (n+1)\,p\rfloor & \text{if }(n+1)p\text{ is 0 or a noninteger}, \\

(n+1)\,p\ \text{ and }\ (n+1)\,p - 1 &\text{if }(n+1)p\in\{1,\dots,n\}, \\

n & \text{if }(n+1)p = n + 1.

\end{cases}</math>

Proof: Let

For only has a nonzero value with . For we find and for . This proves that the mode is 0 for and for .

Let . We find

From this follows

<math display="block">\begin{align}

k > (n+1)p-1 \Rightarrow f(k+1) < f(k) \\

k = (n+1)p-1 \Rightarrow f(k+1) = f(k) \\

k < (n+1)p-1 \Rightarrow f(k+1) > f(k)

\end{align}</math>

So when is an integer, then and is a mode. In the case that , then only is a mode.See also

Median

In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However, several special results have been established:

  • If is an integer, then the mean, median, and mode coincide and equal .Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", The Mathematical Gazette 94, 331-332.
  • Any median must lie within the interval .
  • A median cannot lie too far away from the mean:.<ref name="Hamza">

{{cite journal

| last1 = Hamza | first1 = K.

| doi = 10.1016/0167-7152(94)00090-U

| title = The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions

| journal = Statistics & Probability Letters

| volume = 23

| pages = 21–25

| year = 1995

}}</ref>

  • The median is unique and equal to when (except for the case when and is odd).
  • When is a rational number (with the exception of and odd), the median is unique.<ref name="Nowakowski">

{{cite journal

| last1 = Nowakowski | first1 = Sz.

| doi = 10.37418/amsj.10.4.9

| issn=1857-8365

| title = Uniqueness of a Median of a Binomial Distribution with Rational Probability

| journal = Advances in Mathematics: Scientific Journal

| volume = 10

| issue = 4

| pages = 1951–1958

| year = 2021

| arxiv = 2004.03280

| s2cid = 215238991

}}</ref>

  • When and is odd, any number in the interval is a median of the binomial distribution. If and is even, then is the unique median.

Tail bounds

For , upper bounds can be derived for the lower tail of the cumulative distribution function , the probability that there are at most successes. Since , these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for .

Hoeffding's inequality yields the simple bound

which is however not very tight. In particular, for , we have that (for fixed , with ), but Hoeffding's bound evaluates to a positive constant.

A sharper bound can be obtained from the Chernoff bound:

where is the relative entropy (or Kullback-Leibler divergence) between an -coin and a -coin (that is, between the and distribution):

Asymptotically, this bound is reasonably tight; see for details.

One can also obtain lower bounds on the tail , known as anti-concentration bounds. By approximating the binomial coefficient with Stirling's formula it can be shown that

which implies the simpler but looser bound

For and for even , it is possible to make the denominator constant:

Statistical inference

Estimation of parameters

When is known, the parameter can be estimated using the proportion of successes:

This estimator is found using maximum likelihood estimator and also the method of moments. This estimator is unbiased and uniformly with minimum variance, proven using Lehmann–Scheffé theorem, since it is based on a minimal sufficient and complete statistic (that is, ). It is also consistent both in probability and in MSE. This statistic is asymptotically normal thanks to the central limit theorem, because it is the same as taking the mean over Bernoulli samples. It has a variance of , a property which is used in various ways, such as in Wald's confidence intervals.

A closed form Bayes estimator for also exists when using the Beta distribution as a conjugate prior distribution. When using a general as a prior, the posterior mean estimator is:

The Bayes estimator is asymptotically efficient and as the sample size approaches infinity (), it approaches the MLE solution. The Bayes estimator is biased (how much depends on the priors), admissible and consistent in probability. Using the Bayesian estimator with the Beta distribution can be used with Thompson sampling.

For the special case of using the standard uniform distribution as a non-informative prior, , the posterior mean estimator becomes:

(A posterior mode should just lead to the standard estimator.) This method is called the rule of succession, which was introduced in the 18th century by Pierre-Simon Laplace.

When relying on Jeffreys prior, the prior is ,Marko Lalovic (https://stats.stackexchange.com/users/105848/marko-lalovic), Jeffreys prior for binomial likelihood, URL (version: 2019-03-04): https://stats.stackexchange.com/q/275608 which leads to the estimator:

When estimating with very rare events and a small (for example, if ), 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 obtained using the rule of three:

Confidence intervals for the parameter p

Even for quite large values of , the actual distribution of the mean is significantly nonnormal. Because of this problem several methods to estimate confidence intervals have been proposed.

In the equations for confidence intervals below, the variables have the following meaning:

  • is the number of successes out of , the total number of trials
  • is the proportion of successes
  • is the quantile of a standard normal distribution (that is, probit) corresponding to the target error rate . For example, for a 95% confidence level the error , so and .

Wald method

A continuity correction of may be added.

Agresti–Coull method

Here the estimate of is modified to

This method works well for and . See here for . For use the Wilson (score) method below.

Arcsine method

Wilson (score) method

The notation in the formula below differs from the previous formulas in two respects:

  • Firstly, has a slightly different interpretation in the formula below: it has its ordinary meaning of 'the th quantile of the standard normal distribution', rather than being a shorthand for 'the th quantile'.
  • Secondly, this formula does not use a plus-minus to define the two bounds. Instead, one may use to get the lower bound, or use to get the upper bound. For example: for a 95% confidence level the error , so one gets the lower bound by using , and one gets the upper bound by using .

<math display="block">\frac{

\hat{p} + \frac{z^2}{2n} + z

\sqrt{

\frac{\hat{p} \left(1 - \hat{p}\right)}{n} +

\frac{z^2}{4 n^2}

}

}{

1 + \frac{z^2}{n}

}</math><ref>

{{cite book

| chapter = Confidence intervals

| chapter-url = http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm

| title = Engineering Statistics Handbook

| publisher = NIST/Sematech

| year = 2012

| access-date = 2017-07-23

}}</ref>

Comparison

The so-called "exact" (Clopper–Pearson) method is the most conservative. (Exact does not mean perfectly accurate; rather, it indicates that the estimates will not be less conservative than the true value.)

The Wald method, although commonly recommended in textbooks, is the most biased.

Sums of binomials

If and are independent binomial variables with the same probability , then is again a binomial variable; its distribution is :

<math display="block">\begin{align}

\operatorname P(Z=k) &= \sum_{i=0}^k\left[\binom{n}i p^i (1-p)^{n-i}\right]\left[\binom{m}{k-i} p^{k-i} (1-p)^{m-k+i}\right]\\

&= \binom{n+m}k p^k (1-p)^{n+m-k}

\end{align}</math>

A Binomial distributed random variable can be considered as the sum of Bernoulli distributed random variables. So the sum of two Binomial distributed random variables and is equivalent to the sum of Bernoulli distributed random variables, which means . This can also be proven directly using the addition rule.

However, if and do not have the same probability , then the variance of the sum will be smaller than the variance of a binomial variable distributed as .

Poisson binomial distribution

The binomial distribution is a special case of the Poisson binomial distribution, which is the distribution of a sum of independent non-identical Bernoulli trials .<ref>

{{cite journal

| volume = 3

| issue = 2

| pages = 295–312

| last = Wang

| first = Y. H.

| title = On the number of successes in independent trials

| journal = Statistica Sinica

| year = 1993

| url = http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf

| archive-url = https://web.archive.org/web/20160303182353/http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf

| archive-date = 2016-03-03

}}

</ref>

Ratio of two binomial distributions

This result was first derived by Katz and coauthors in 1978.

Let and be independent. Let .

Then is approximately normally distributed with mean and variance .

Conditional binomials

If and (the conditional distribution of , given ), then is a simple binomial random variable with distribution .

For example, imagine throwing balls to a basket and taking the balls that hit and throwing them to another basket . If is the probability to hit then is the number of balls that hit . If is the probability to hit then the number of balls that hit is and therefore .

Since and , by the law of total probability,

<math display="block">\begin{align}

\Pr[Y = m] &= \sum_{k = m}^{n} \Pr[Y = m \mid X = k] \Pr[X = k] \\[2pt]

&= \sum_{k=m}^n \binom{n}{k} \binom{k}{m} p^k q^m (1-p)^{n-k} (1-q)^{k-m}

\end{align}</math>

Since the equation above can be expressed as

Factoring and pulling all the terms that don't depend on out of the sum now yields

<math display="block">\begin{align}

\Pr[Y = m] &= \binom{n}{m} p^m q^m \left( \sum_{k=m}^n \binom{n-m}{k-m} p^{k-m} (1-p)^{n-k} (1-q)^{k-m} \right) \\[2pt]

&= \binom{n}{m} (pq)^m \left( \sum_{k=m}^n \binom{n-m}{k-m} \left(p(1-q)\right)^{k-m} (1-p)^{n-k} \right)

\end{align}</math>

After substituting in the expression above, we get

Notice that the sum (in the parentheses) above equals by the binomial theorem. Substituting this in finally yields

<math display="block">\begin{align}

\Pr[Y=m] &= \binom{n}{m} (pq)^m (p - pq + 1 - p)^{n-m}\\[4pt]

&= \binom{n}{m} (pq)^m (1-pq)^{n-m}

\end{align}</math>

and thus as desired.

Bernoulli distribution

The Bernoulli distribution is a special case of the binomial distribution, where . Symbolically, has the same meaning as . Conversely, any binomial distribution, , is the distribution of the sum of independent Bernoulli trials, , each with the same probability .

Normal approximation

right|250px|thumb|Binomial probability mass function and normal probability density function approximation for and

If is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to 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 increases (at least 20) and is better when is not near to 0 or 1. Various rules of thumb may be used to decide whether is large enough, and is far enough from the extremes of zero or one:

  • One rule is that for the 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.

  • A stronger rule states 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, only if

This 3-standard-deviation rule is equivalent to the following conditions, which also imply the first rule above.

The rule is totally equivalent to request that

Moving terms around yields:

Since , we can apply the square power and divide by the respective factors and , to obtain the desired conditions:

Notice that these conditions automatically imply that . On the other hand, apply again the square root and divide by 3,

Subtracting the second set of inequalities from the first one yields:

and so, the desired first rule is satisfied,

  • Another commonly used rule is that both values and must be greater than or equal to 5. However, the specific number varies from source to source, and depends on how good an approximation one wants. In particular, if one uses 9 instead of 5, the rule implies the results stated in the previous paragraphs.

Assume that both values and are greater than 9. Since , we easily have that

We only have to divide now by the respective factors and , to deduce the alternative form of the 3-standard-deviation rule:

The following is an example of applying a continuity correction. Suppose one wishes to calculate for a binomial random variable . If has a distribution given by the normal approximation, then is approximated by . 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 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 1738. Nowadays, it can be seen as a consequence of the central limit theorem since is a sum of independent, identically distributed Bernoulli variables with parameter . This fact is the basis of a hypothesis test, a "proportion z-test", for the value of using , the sample proportion and estimator of , in a common test statistic.NIST/SEMATECH, "7.2.4. Does the proportion of defectives meet requirements?" e-Handbook of Statistical Methods.

For example, suppose one randomly samples 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 people were sampled repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion of agreement in the population and with standard deviation

Poisson approximation

The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product converges to a finite limit. Therefore, the Poisson distribution with parameter can be used as an approximation to of the binomial distribution if is sufficiently large and is sufficiently small. According to rules of thumb, this approximation is good if and such that , or if and such that , or if and .NIST/SEMATECH, "6.3.3.1. Counts Control Charts", e-Handbook of Statistical Methods.

Concerning the accuracy of Poisson approximation, see Novak,Novak S.Y. (2011) Extreme value methods with applications to finance. London: CRC/ Chapman & Hall/Taylor & Francis. . ch. 4, and references therein.

Limiting distributions

  • Poisson limit theorem: As approaches and approaches 0 with the product held fixed, the distribution approaches the Poisson distribution with expected value .
  • de Moivre–Laplace theorem: As approaches while 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 is asymptotically normal with expected value 0 and variance 1. This result is a specific case of the central limit theorem.

Beta distribution

The binomial distribution and beta distribution are different views of the same model of repeated Bernoulli trials. The binomial distribution is the PMF of successes given independent events each with a probability of success.

Mathematically, when and , the beta distribution and the binomial distribution are related by{{clarification needed|date=March 2023|

reason=Is the left hand side referring to a probability density, and the right hand side to a probability mass function? Clearly a beta distributed random variable can not be a scalar multiple of a binomial random variable given that the former is continuous and the latter discrete. In any case, it would seem to be more correct to say that this relationship means that the PDF of one is related to the PMF of the other, rather than appearing to say that the _distributions_ (often interchangeable with their CDFs) are directly related to one another.

}} a factor of :

Beta distributions also provide a family of prior probability distributions for binomial distributions in Bayesian inference:

Given a uniform prior, the posterior distribution for the probability of success given independent events with observed successes is a beta distribution.

Computational methods

Random number generation

Methods for random number generation where the marginal distribution is a binomial distribution are well-established.Devroye, Luc (1986) Non-Uniform Random Variate Generation, New York: Springer-Verlag. (See especially Chapter X, Discrete Univariate Distributions)<ref>

{{cite journal

| pages = 216–222

| year = 1988

| doi = 10.1145/42372.42381

| last2 = Schmeiser| first1 = V.

| volume = 31| first2 = B. W.

| journal = Communications of the ACM

| title = Binomial random variate generation

| last1 = Kachitvichyanukul| issue = 2

| s2cid = 18698828

}}</ref>

One way to generate random variates samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that for all values from through . (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 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.

History

This distribution was derived by Jacob Bernoulli. He considered the case where where is the probability of success and and are positive integers. Blaise Pascal had earlier considered the case where , tabulating the corresponding binomial coefficients in what is now recognized as Pascal's triangle.

See also

  • Logistic regression
  • Multinomial distribution
  • Negative binomial distribution
  • Beta-binomial distribution
  • Binomial measure, an example of a multifractal measure.Mandelbrot, B. B., Fisher, A. J., & Calvet, L. E. (1997). A multifractal model of asset returns. 3.2 The Binomial Measure is the Simplest Example of a Multifractal
  • Statistical mechanics
  • Piling-up lemma, the resulting probability when XOR-ing independent Boolean variables

Notes

References

Further reading

  • Interactive graphic: Univariate Distribution Relationships
  • Difference of two binomial variables: X-Y or |X-Y|
  • Querying the binomial probability distribution in WolframAlpha
  • Confidence (credible) intervals for binomial probability, p: online calculator available at causaScientia.org