Binomial distribution#Covariance between two binomials

If the random variable {{mvar|X}} follows the binomial distribution with parameters {{math|n ∈ $\backslash mathbb\{N\}$}} and {{math|p ∈ {{closed-closed|0, 1}}}}, we write {{math|X ~ B(n, p)}}. The probability of getting exactly {{mvar|k}} successes in {{mvar|n}} independent Bernoulli trials (with the same rate {{mvar|p}}) is given by the probability mass function:

: $f(k,n,p)\; =\; \backslash Pr(X\; =\; k)\; =\; \backslash binom\{n\}\{k\}p^k(1-p)^\{n-k\}$

for {{math|k {{=}} 0, 1, 2, ..., n}}, where

: $\backslash binom\{n\}\{k\}\; =\backslash frac\{n!\}\{k!(n-k)!\}$

is the binomial coefficient. The formula can be understood as follows: {{math|p{{sup|k}} q{{sup|n−k}}}} is the probability of obtaining the sequence of {{mvar|n}} independent Bernoulli trials in which {{mvar|k}} trials are "successes" and the remaining {{math|n − k}} trials result in "failure". Since the trials are independent with probabilities remaining constant between them, any sequence of {{mvar|n}} trials with {{mvar|k}} successes (and {{math|n − k}} failures) has the same probability of being achieved (regardless of positions of successes within the sequence). There are $\backslash binom\{n\}\{k\}$ such sequences, since the binomial coefficient $\backslash binom\{n\}\{k\}$ counts the number of ways to choose the positions of the {{mvar|k}} successes among the {{mvar|n}} trials. The binomial distribution is concerned with the probability of obtaining any of these sequences, meaning the probability of obtaining one of them ({{math|p{{sup|k}} q{{sup|n−k}}}}) must be added $\backslash binom\{n\}\{k\}$ times, hence $\backslash Pr(X\; =\; k)\; =\; \backslash binom\{n\}\{k\}\; p^k\; (1-p)^\{n-k\}$.

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

: $f(k,n,p)=f(n-k,n,1-p).$

Looking at the expression {{math|f(k, n, p)}} as a function of {{mvar|k}}, there is a {{mvar|k}} value that maximizes it. This {{mvar|k}} value can be found by calculating

: $\backslash frac\{f(k+1,n,p)\}\{f(k,n,p)\}=\backslash frac\{(n-k)p\}\{(k+1)(1-p)\}$

and comparing it to 1. There is always an integer {{mvar|M}} that satisfies{{cite book |last=Feller |first=W. |title=An Introduction to Probability Theory and Its Applications |url=https://archive.org/details/introductiontopr01wfel |url-access=limited |year=1968 |publisher=Wiley |location=New York |edition=Third |page=[https://archive.org/details/introductiontopr01wfel/page/n167 151] (theorem in section VI.3) }}

:

{{math|f(k, n, p)}} is monotone increasing for {{math|k < M}} and monotone decreasing for {{math|k > M}}, with the exception of the case where {{math|(n + 1)p}} is an integer. In this case, there are two values for which {{mvar|f}} is maximal: {{math|(n + 1) p}} and {{math|(n + 1) p − 1}}. {{mvar|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.

Equivalently, {{math|M − p < np ≤ M + 1 − p}}. Taking the floor function, we obtain {{math|M {{=}} floor(np)}}.{{NoteTag|Except the trivial case {{math|p {{=}} 0}}, which must be checked separately.}}

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

: $f(4,6,0.3)\; =\; \backslash binom\{6\}\{4\}0.3^4\; (1-0.3)^\{6-4\}=\; 0.059535.$

The cumulative distribution function can be expressed as:

: $F(k;n,p)\; =\; \backslash Pr(X\; \backslash le\; k)\; =\; \backslash sum\_\{i=0\}^\{\backslash lfloor\; k\; \backslash rfloor\}\; \{n\backslash choose\; i\}p^i(1-p)^\{n-i\},$

where $\backslash lfloor\; k\backslash rfloor$ is the "floor" under {{math|k}}, i.e. the greatest integer less than or equal to {{math|k}}.

It can also be represented in terms of the regularized incomplete beta function, as follows:{{cite book |last=Wadsworth |first=G. P. |title=Introduction to Probability and Random Variables |year=1960 |publisher=McGraw-Hill |location=New York |page=[https://archive.org/details/introductiontopr0000wads/page/52 52] |url=https://archive.org/details/introductiontopr0000wads |url-access=registration }}

: $\backslash begin\{align\}$

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}

which is equivalent to the cumulative distribution functions of the beta distribution and of the F-distribution:{{cite journal |last=Jowett |first=G. H. |year=1963 |title=The Relationship Between the Binomial and F Distributions |journal=Journal of the Royal Statistical Society, Series D |volume=13 |issue=1 |pages=55–57 |doi=10.2307/2986663 |jstor=2986663 }}

: $F(k;n,p)\; =\; F\_\{\backslash text\{beta-distribution\}\}\backslash left(x=1-p;\backslash alpha=n-k,\backslash beta=k+1\backslash right)$

: $F(k;n,p)\; =\; F\_\{F\backslash text\{-distribution\}\}\backslash left(x=\backslash frac\{1-p\}\{p\}\backslash frac\{k+1\}\{n-k\};d\_1=2(n-k),d\_2=2(k+1)\backslash right).$

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

If {{math|X ~ B(n, p)}}, that is, {{math|X}} is a binomially distributed random variable, {{mvar|n}} being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of {{math|X}} is:See [https://proofwiki.org/wiki/Expectation_of_Binomial_Distribution Proof Wiki]

: $\backslash operatorname\{E\}[X]\; =\; np.$

This follows from the linearity of the expected value along with the fact that {{mvar|X}} is the sum of {{mvar|n}} identical Bernoulli random variables, each with expected value {{mvar|p}}. In other words, if $X\_1,\; \backslash ldots,\; X\_n$ are identical (and independent) Bernoulli random variables with parameter {{mvar|p}}, then {{math|1=X = X₁ + ... + X_n}} and

: $\backslash operatorname\{E\}[X]\; =\; \backslash operatorname\{E\}[X\_1\; +\; \backslash cdots\; +\; X\_n]\; =\; \backslash operatorname\{E\}[X\_1]\; +\; \backslash cdots\; +\; \backslash operatorname\{E\}[X\_n]\; =\; p\; +\; \backslash cdots\; +\; p\; =\; np.$

The variance is:

: $\backslash operatorname\{Var\}(X)\; =\; npq\; =\; np(1\; -\; p).$

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

The first 6 central moments, defined as $\backslash mu\; \_\{c\}=\backslash operatorname\; \{E\}\; \backslash left[(X-\backslash operatorname\; \{E\}\; [X])^\{c\}\backslash right]$, are given by

: $\backslash begin\{align\}$

\mu_1 &= 0, \\

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

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

\mu_4 &= np(1-p)(1+(3n-6)p(1-p)),\\

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

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

\end{align}

The non-central moments satisfy

: $\backslash begin\{align\}$

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

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

\end{align}

and in general

{{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

|url=https://www.jstor.org/stable/40233780

|volume=69

|issue=1

|pages=197–204

|doi=10.1137/070700024

|jstor=40233780

}}

{{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

}}

: $$

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

where $\backslash textstyle\; \backslash left\backslash \{\{c\backslash atop\; k\}\backslash right\backslash \}$ are the Stirling numbers of the second kind, and $n^\{\backslash underline\{k\}\}\; =\; n(n-1)\backslash cdots(n-k+1)$ is the $k$th falling power of $n$.

A simple bound

{{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

|page=109306 |arxiv=2103.17027 }} follows by bounding the Binomial moments via the higher Poisson moments:

: $$

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

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

This shows that if $c=O(\backslash sqrt\{np\})$, then $\backslash operatorname\; \{E\}[X^c]$ is at most a constant factor away from $\backslash operatorname\; \{E\}[X]^c$

Usually the mode of a binomial {{math|B(n, p)}} distribution is equal to $\backslash lfloor\; (n+1)p\backslash rfloor$, where $\backslash lfloor\backslash cdot\backslash rfloor$ is the floor function. However, when {{math|(n + 1)p}} is an integer and {{math|p}} is neither 0 nor 1, then the distribution has two modes: {{math|(n + 1)p}} and {{math|(n + 1)p − 1}}. When {{math|p}} is equal to 0 or 1, the mode will be 0 and {{math|n}} correspondingly. These cases can be summarized as follows:

: $\backslash text\{mode\}\; =$

\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}

Proof: Let

: $f(k)=\backslash binom\; nk\; p^k\; q^\{n-k\}.$

For $p=0$ only $f(0)$ has a nonzero value with $f(0)=1$. For $p=1$ we find $f(n)=1$ and $f(k)=0$ for $k\backslash neq\; n$. This proves that the mode is 0 for $p=0$ and $n$ for $p=1$.

Let . We find

:$\backslash frac\{f(k+1)\}\{f(k)\}\; =\; \backslash frac\{(n-k)p\}\{(k+1)(1-p)\}$.

From this follows

: $\backslash begin\{align\}$

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}

So when $(n+1)p-1$ is an integer, then $(n+1)p-1$ and $(n+1)p$ is a mode. In the case that $(n+1)p-1\backslash notin\; \backslash Z$, then only $\backslash lfloor\; (n+1)p-1\backslash rfloor+1=\backslash lfloor\; (n+1)p\backslash rfloor$ is a mode.See also {{cite web |first=André |last=Nicolas |title=Finding mode in Binomial distribution |work=Stack Exchange |date=January 7, 2019 |url=https://math.stackexchange.com/q/117940 }}

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 {{math|np}} is an integer, then the mean, median, and mode coincide and equal {{math|np}}.{{cite journal|last=Neumann|first=P.|year=1966|title=Über den Median der Binomial- and Poissonverteilung|journal=Wissenschaftliche Zeitschrift der Technischen Universität Dresden|volume=19|pages=29–33|language=de}}Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", The Mathematical Gazette 94, 331-332.
• Any median {{math|m}} must lie within the interval $\backslash lfloor\; np\; \backslash rfloor\backslash leq\; m\; \backslash leq\; \backslash lceil\; np\; \backslash rceil$.{{cite journal|first1=R.|last1=Kaas|first2=J.M.|last2=Buhrman|title=Mean, Median and Mode in Binomial Distributions|journal=Statistica Neerlandica|year=1980|volume=34|issue=1|pages=13–18|doi=10.1111/j.1467-9574.1980.tb00681.x}}
• A median {{math|m}} cannot lie too far away from the mean:$|m-np|\backslash leq\; \backslash min\backslash \{\{\backslash ln2\},\; \backslash max\backslash \{p,1-p\backslash \}\backslash \}$ .

{{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

}}

• The median is unique and equal to {{math|1=m = round(np)}} when {{math|1={{abs|m − np}} ≤ min{{brace|p, 1 − p}}}} (except for the case when {{math|1=p = 1/2}} and {{math|n}} is odd).
• When {{math|p}} is a rational number (with the exception of {{math|1=p = 1/2}}\ and {{math|n}} odd) the median is unique.

{{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

}}

• When $p=\; \backslash frac\{1\}\{2\}$ and {{math|n}} is odd, any number {{math|m}} in the interval $\backslash frac\{1\}\{2\}\; \backslash bigl(n-1\backslash bigr)\backslash leq\; m\; \backslash leq\; \backslash frac\{1\}\{2\}\; \backslash bigl(n+1\backslash bigr)$ is a median of the binomial distribution. If $p=\; \backslash frac\{1\}\{2\}$ and {{math|n}} is even, then $m=\; \backslash frac\{n\}\{2\}$ is the unique median.

For {{math|k ≤ np}}, upper bounds can be derived for the lower tail of the cumulative distribution function $F(k;n,p)\; =\; \backslash Pr(X\; \backslash le\; k)$, the probability that there are at most {{math|k}} successes. Since $\backslash Pr(X\; \backslash ge\; k)\; =\; F(n-k;n,1-p)$, these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for {{math|k ≥ np}}.

Hoeffding's inequality yields the simple bound

: $F(k;n,p)\; \backslash leq\; \backslash exp\backslash left(-2\; n\backslash left(p-\backslash frac\{k\}\{n\}\backslash right)^2\backslash right),\; \backslash !$

which is however not very tight. In particular, for {{math|1=p = 1}}, we have that {{math|1=F(k; n, p) = 0}} (for fixed {{math|k}}, {{math|n}} with {{math|k < n}}), but Hoeffding's bound evaluates to a positive constant.

A sharper bound can be obtained from the Chernoff bound:{{cite journal |first1=R. |last1=Arratia |first2=L. |last2=Gordon |title=Tutorial on large deviations for the binomial distribution |journal=Bulletin of Mathematical Biology |volume=51 |issue=1 |year=1989 |pages=125–131 |doi=10.1007/BF02458840 |pmid=2706397 |s2cid=189884382 }}

: $F(k;n,p)\; \backslash leq\; \backslash exp\backslash left(-nD\backslash left(\backslash frac\{k\}\{n\}\backslash parallel\; p\backslash right)\backslash right)$

where {{math|D(a ∥ p)}} is the relative entropy (or Kullback-Leibler divergence) between an {{math|a}}-coin and a {{math|p}}-coin (i.e. between the {{math|Bernoulli(a)}} and {{math|Bernoulli(p)}} distribution):

: $D(a\backslash parallel\; p)=(a)\backslash ln\backslash frac\{a\}\{p\}+(1-a)\backslash ln\backslash frac\{1-a\}\{1-p\}.\; \backslash !$

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

One can also obtain lower bounds on the tail {{math|F(k; n, p)}}, known as anti-concentration bounds. By approximating the binomial coefficient with Stirling's formula it can be shown that{{cite book |author1=Robert B. Ash |title=Information Theory |url=https://archive.org/details/informationtheor00ashr |url-access=limited |date=1990 |publisher=Dover Publications |page=[https://archive.org/details/informationtheor00ashr/page/n81 115]|isbn=9780486665214 }}

: $F(k;n,p)\; \backslash geq\; \backslash frac\{1\}\{\backslash sqrt\{8n\backslash tfrac\{k\}\{n\}(1-\backslash tfrac\{k\}\{n\})\}\}\; \backslash exp\backslash left(-nD\backslash left(\backslash frac\{k\}\{n\}\backslash parallel\; p\backslash right)\backslash right),$

which implies the simpler but looser bound

: $F(k;n,p)\; \backslash geq\; \backslash frac1\{\backslash sqrt\{2n\}\}\; \backslash exp\backslash left(-nD\backslash left(\backslash frac\{k\}\{n\}\backslash parallel\; p\backslash right)\backslash right).$

For {{math|1=p = 1/2}} and {{math|k ≥ 3n/8}} for even {{math|n}}, it is possible to make the denominator constant:{{cite web |last1=Matoušek |first1=J. |last2=Vondrak |first2=J. |title=The Probabilistic Method |work=lecture notes |url=https://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15859-f09/www/handouts/matousek-vondrak-prob-ln.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15859-f09/www/handouts/matousek-vondrak-prob-ln.pdf |archive-date=2022-10-09 |url-status=live }}

: $F(k;n,\backslash tfrac\{1\}\{2\})\; \backslash geq\; \backslash frac\{1\}\{15\}\; \backslash exp\backslash left(-\; 16n\; \backslash left(\backslash frac\{1\}\{2\}\; -\backslash frac\{k\}\{n\}\backslash right)^2\backslash right).\; \backslash !$

{{see also|Beta distribution#Bayesian inference}}

When {{math|n}} is known, the parameter {{math|p}} can be estimated using the proportion of successes:

: $\backslash widehat\{p\}\; =\; \backslash frac\{x\}\{n\}.$

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 (i.e.: {{math|x}}). 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 $var(\backslash widehat\{p\})\; =\; \backslash frac\{p(1-p)\}\{n\}$, a property which is used in various ways, such as in Wald's confidence intervals.

A closed form Bayes estimator for {{math|p}} also exists when using the Beta distribution as a conjugate prior distribution. When using a general $\backslash operatorname\{Beta\}(\backslash alpha,\; \backslash beta)$ as a prior, the posterior mean estimator is:

: $\backslash widehat\{p\}\_b\; =\; \backslash frac\{x+\backslash alpha\}\{n+\backslash alpha+\backslash beta\}.$

The Bayes estimator is asymptotically efficient and as the sample size approaches infinity ({{math|n → ∞}}), it approaches the MLE solution.{{Cite journal |last=Wilcox |first=Rand R. |date=1979 |title=Estimating the Parameters of the Beta-Binomial Distribution |url=http://journals.sagepub.com/doi/10.1177/001316447903900302 |journal=Educational and Psychological Measurement |language=en |volume=39 |issue=3 |pages=527–535 |doi=10.1177/001316447903900302 |s2cid=121331083 |issn=0013-1644}} 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, $\backslash operatorname\{Beta\}(\backslash alpha=1,\; \backslash beta=1)\; =\; U(0,1)$, the posterior mean estimator becomes:

:$\backslash widehat\{p\}\_b\; =\; \backslash frac\{x+1\}\{n+2\}.$

(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 $\backslash operatorname\{Beta\}(\backslash alpha=\backslash frac\{1\}\{2\},\; \backslash beta=\backslash frac\{1\}\{2\})$,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:

: $\backslash widehat\{p\}\_\{Jeffreys\}\; =\; \backslash frac\{x+\backslash frac\{1\}\{2\}\}\{n+1\}.$

When estimating {{math|p}} with very rare events and a small {{math|n}} (e.g.: if {{math|1=x = 0}}), then using the standard estimator leads to $\backslash widehat\{p\}\; =\; 0,$ which sometimes is unrealistic and undesirable. In such cases there are various alternative estimators.{{cite journal |last=Razzaghi |first=Mehdi |title=On the estimation of binomial success probability with zero occurrence in sample |journal=Journal of Modern Applied Statistical Methods |volume=1 |issue=2 |year=2002 |pages=326–332 |doi=10.22237/jmasm/1036110000 |doi-access=free }} One way is to use the Bayes estimator $\backslash widehat\{p\}\_b$, leading to:

: $\backslash widehat\{p\}\_b\; =\; \backslash frac\{1\}\{n+2\}.$

Another method is to use the upper bound of the confidence interval obtained using the rule of three:

: $\backslash widehat\{p\}\_\{\backslash text\{rule\; of\; 3\}\}\; =\; \backslash frac\{3\}\{n\}.$

{{Main|Binomial proportion confidence interval}}

{{see also|Z-test#Comparing the Proportions of Two Binomials}}

Even for quite large values of n, the actual distribution of the mean is significantly nonnormal.{{Citation |first1=Lawrence D. |last1=Brown |first2=T. Tony |last2=Cai |first3=Anirban |last3=DasGupta |year=2001 |title = Interval Estimation for a Binomial Proportion |url=http://www-stat.wharton.upenn.edu/~tcai/paper/html/Binomial-StatSci.html |journal=Statistical Science |volume=16 |issue=2 |pages=101–133 |access-date = 2015-01-05 |doi=10.1214/ss/1009213286|citeseerx=10.1.1.323.7752 }} 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:

• n₁ is the number of successes out of n, the total number of trials
• $\backslash widehat\{p\backslash ,\}\; =\; \backslash frac\{n\_1\}\{n\}$ is the proportion of successes
• $z$ is the $1\; -\; \backslash tfrac\{1\}\{2\}\backslash alpha$ quantile of a standard normal distribution (i.e., probit) corresponding to the target error rate $\backslash alpha$. For example, for a 95% confidence level the error $\backslash alpha$ = 0.05, so $1\; -\; \backslash tfrac\{1\}\{2\}\backslash alpha$ = 0.975 and $z$ = 1.96.

{{Main|Binomial proportion confidence interval#Wald interval}}

: $\backslash widehat\{p\backslash ,\}\; \backslash pm\; z\; \backslash sqrt\{\; \backslash frac\{\; \backslash widehat\{p\backslash ,\}\; (\; 1\; -\backslash widehat\{p\backslash ,\}\; )\}\{\; n\; \}\; \}\; .$

A continuity correction of {{math|0.5/n}} may be added.{{clarify|date=July 2012}}

{{Main|Binomial proportion confidence interval#Agresti–Coull interval}}

{{Citation |last1=Agresti |first1=Alan |last2=Coull |first2=Brent A. |date=May 1998 |title=Approximate is better than 'exact' for interval estimation of binomial proportions |url = http://www.stat.ufl.edu/~aa/articles/agresti_coull_1998.pdf |journal=The American Statistician |volume=52 |issue=2 |pages=119–126 |access-date=2015-01-05 |doi=10.2307/2685469 |jstor=2685469 }}

: $\backslash tilde\{p\}\; \backslash pm\; z\; \backslash sqrt\{\; \backslash frac\{\; \backslash tilde\{p\}\; (\; 1\; -\; \backslash tilde\{p\}\; )\}\{\; n\; +\; z^2\; \}\; \}$

Here the estimate of {{math|p}} is modified to

: $\backslash tilde\{p\}=\; \backslash frac\{\; n\_1\; +\; \backslash frac\{1\}\{2\}\; z^2\}\{\; n\; +\; z^2\; \}$

This method works well for {{math|n > 10}} and {{math|n₁ ≠ 0, n}}.{{cite web|last1=Gulotta|first1=Joseph|title=Agresti-Coull Interval Method|url=https://pellucid.atlassian.net/wiki/spaces/PEL/pages/25722894/Agresti-Coull+Interval+Method#:~:text=The%20Agresti%2DCoull%20Interval%20Method,%2C%20or%20per%20100%2C000%2C%20etc|website=pellucid.atlassian.net|access-date=18 May 2021}} See here for $n\backslash leq\; 10$.{{cite web|title=Confidence intervals|url=https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm|website=itl.nist.gov|access-date=18 May 2021}} For {{math|1=n₁ = 0, n}} use the Wilson (score) method below.

{{Main|Binomial proportion confidence interval#Arcsine transformation}}

{{cite book |last=Pires |first=M. A. |chapter-url=https://www.math.tecnico.ulisboa.pt/~apires/PDFs/AP_COMPSTAT02.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.math.tecnico.ulisboa.pt/~apires/PDFs/AP_COMPSTAT02.pdf |archive-date=2022-10-09 |url-status=live |chapter=Confidence intervals for a binomial proportion: comparison of methods and software evaluation |editor-last=Klinke |editor-first=S. |editor2-last=Ahrend |editor2-first=P. |editor3-last=Richter |editor3-first=L. |title=Proceedings of the Conference CompStat 2002 |others=Short Communications and Posters |year=2002 }}

: $\backslash sin^2\; \backslash left(\backslash arcsin\; \backslash left(\backslash sqrt\{\backslash widehat\{p\backslash ,\}\}\backslash right)\; \backslash pm\; \backslash frac\{z\}\{2\backslash sqrt\{n\}\}\; \backslash right).$

{{Main|Binomial proportion confidence interval#Wilson score interval}}

The notation in the formula below differs from the previous formulas in two respects:{{Citation |last = Wilson |first=Edwin B. |date = June 1927 |title = Probable inference, the law of succession, and statistical inference |url = http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf |journal = Journal of the American Statistical Association |volume=22 |issue=158 |pages=209–212 |access-date= 2015-01-05 |doi = 10.2307/2276774 |url-status=dead |archive-url = https://web.archive.org/web/20150113082307/http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf |archive-date = 2015-01-13 |jstor = 2276774 }}

• Firstly, {{math|z_x}} has a slightly different interpretation in the formula below: it has its ordinary meaning of 'the {{math|x}}th quantile of the standard normal distribution', rather than being a shorthand for 'the {{math|(1 − x)}}th quantile'.
• Secondly, this formula does not use a plus-minus to define the two bounds. Instead, one may use $z\; =\; z\_\{\backslash alpha\; /\; 2\}$ to get the lower bound, or use $z\; =\; z\_\{1\; -\; \backslash alpha/2\}$ to get the upper bound. For example: for a 95% confidence level the error $\backslash alpha$ = 0.05, so one gets the lower bound by using $z\; =\; z\_\{\backslash alpha/2\}\; =\; z\_\{0.025\}\; =\; -\; 1.96$, and one gets the upper bound by using $z\; =\; z\_\{1\; -\; \backslash alpha/2\}\; =\; z\_\{0.975\}\; =\; 1.96$.

: $\backslash frac\{$

\widehat{p\,} + \frac{z^2}{2n} + z

\sqrt{

\frac{\widehat{p\,}(1 - \widehat{p\,})}{n} +

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

}

}{

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

}

{{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

}}

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.{{clarify|reason=what sense of bias is this|date=July 2012}}

If {{math|X ~ B(n, p)}} and {{math|Y ~ B(m, p)}} are independent binomial variables with the same probability {{math|p}}, then {{math|X + Y}} is again a binomial variable; its distribution is {{math|1=Z = X + Y ~ B(n + m, p)}}:{{cite book |last1=Dekking |first1=F.M. |last2=Kraaikamp |first2=C. |last3=Lopohaa |first3=H.P. |last4=Meester |first4=L.E. |title=A Modern Introduction of Probability and Statistics |date=2005 |publisher=Springer-Verlag London |isbn=978-1-84628-168-6 |edition=1 |url=https://www.springer.com/gp/book/9781852338961}}

: $\backslash begin\{align\}$

\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}

A Binomial distributed random variable {{math|X ~ B(n, p)}} can be considered as the sum of {{math|n}} Bernoulli distributed random variables. So the sum of two Binomial distributed random variables {{math|X ~ B(n, p)}} and {{math|Y ~ B(m, p)}} is equivalent to the sum of {{math|n + m}} Bernoulli distributed random variables, which means {{math|1=Z = X + Y ~ B(n + m, p)}}. This can also be proven directly using the addition rule.

However, if {{math|X}} and {{math|Y}} do not have the same probability {{math|p}}, then the variance of the sum will be smaller than the variance of a binomial variable distributed as {{math|B(n + m, {{overline|p}})}}.

The binomial distribution is a special case of the Poisson binomial distribution, which is the distribution of a sum of {{math|n}} independent non-identical Bernoulli trials {{math|B(p_i)}}.

{{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

| url-status = dead

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

| archive-date = 2016-03-03

}}

This result was first derived by Katz and coauthors in 1978.{{cite journal |last1=Katz |first1=D. |display-authors=1 |first2=J. |last2=Baptista |first3=S. P. |last3=Azen |first4=M. C. |last4=Pike |year=1978 |title=Obtaining confidence intervals for the risk ratio in cohort studies |journal=Biometrics |volume=34 |issue=3 |pages=469–474 |doi=10.2307/2530610 |jstor=2530610 }}

Let {{nowrap|X ~ B(n, p₁)}} and {{nowrap|Y ~ B(m, p₂)}} be independent. Let {{nowrap|1=T = (X/n) / (Y/m)}}.

Then log(T) is approximately normally distributed with mean log(p₁/p₂) and variance {{nowrap|((1/p₁) − 1)/n + ((1/p₂) − 1)/m}}.

If 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).

For example, imagine throwing n balls to a basket U_X and taking the balls that hit and throwing them to another basket U_Y. If p is the probability to hit U_X then X ~ B(n, p) is the number of balls that hit U_X. If q is the probability to hit U_Y then the number of balls that hit U_Y is Y ~ B(X, q) and therefore Y ~ B(n, pq).

{{hidden begin|style=width:60%|ta1=center|border=1px #aaa solid|title=[Proof]}}

Since $X\; \backslash sim\; B(n,\; p)$ and $Y\; \backslash sim\; B(X,\; q)$, by the law of total probability,

: $\backslash begin\{align\}$

\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}

Since $\backslash tbinom\{n\}\{k\}\; \backslash tbinom\{k\}\{m\}\; =\; \backslash tbinom\{n\}\{m\}\; \backslash tbinom\{n-m\}\{k-m\},$ the equation above can be expressed as

: $\backslash Pr[Y\; =\; m]\; =\; \backslash sum\_\{k=m\}^\{n\}\; \backslash binom\{n\}\{m\}\; \backslash binom\{n-m\}\{k-m\}\; p^k\; q^m\; (1-p)^\{n-k\}\; (1-q)^\{k-m\}$

Factoring $p^k\; =\; p^m\; p^\{k-m\}$ and pulling all the terms that don't depend on $k$ out of the sum now yields

: $\backslash begin\{align\}$

\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}

After substituting $i\; =\; k\; -\; m$ in the expression above, we get

: $\backslash Pr[Y\; =\; m]\; =\; \backslash binom\{n\}\{m\}\; (pq)^m\; \backslash left(\; \backslash sum\_\{i=0\}^\{n-m\}\; \backslash binom\{n-m\}\{i\}\; (p\; -\; pq)^i\; (1-p)^\{n-m\; -\; i\}\; \backslash right)$

Notice that the sum (in the parentheses) above equals $(p\; -\; pq\; +\; 1\; -\; p)^\{n-m\}$ by the binomial theorem. Substituting this in finally yields

: $\backslash begin\{align\}$

\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}

and thus $Y\; \backslash sim\; B(n,\; pq)$ as desired.

{{hidden end}}

The Bernoulli distribution is a special case of the binomial distribution, where {{math|1=n = 1}}. Symbolically, {{math|X ~ B(1, p)}} has the same meaning as {{math|X ~ Bernoulli(p)}}. Conversely, any binomial distribution, {{math|B(n, p)}}, is the distribution of the sum of {{math|n}} independent Bernoulli trials, {{math|Bernoulli(p)}}, each with the same probability {{math|p}}.{{cite web|last1=Taboga|first1=Marco|title=Lectures on Probability Theory and Mathematical Statistics|url=https://www.statlect.com/probability-distributions/binomial-distribution#hid3|website=statlect.com|access-date=18 December 2017}}

{{see also|Binomial proportion confidence interval#Normal approximation interval}}

File:Binomial Distribution.svg and normal probability density function approximation for {{math|1=n = 6}} and {{math|1=p = 0.5}}]]

If {{math|n}} is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to {{math|B(n, p)}} is given by the normal distribution

: $\backslash mathcal\{N\}(np,\backslash ,np(1-p)),$

and this basic approximation can be improved in a simple way by using a suitable continuity correction.

The basic approximation generally improves as {{math|n}} increases (at least 20) and is better when {{math|p}} is not near to 0 or 1.{{cite book|title=Statistics for experimenters|url=https://archive.org/details/statisticsforexp00geor|url-access=registration|author=Box, Hunter and Hunter|publisher=Wiley|year=1978|page=[https://archive.org/details/statisticsforexp00geor/page/130 130]|isbn=9780471093152}} Various rules of thumb may be used to decide whether {{math|n}} is large enough, and {{math|p}} is far enough from the extremes of zero or one:

• One rule is that for {{math|n > 5}} the normal approximation is adequate if the absolute value of the skewness is strictly less than 0.3; that is, if
• : $\backslash frac$

  1-2p

  {\sqrt{np(1-p)}}=\frac1{\sqrt{n}}\left|\sqrt{\frac{1-p}p}-\sqrt{\frac{p}{1-p}}\,\right|<0.3.

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
• : $\backslash mu\backslash pm3\backslash sigma=np\backslash pm3\backslash sqrt\{np(1-p)\}\backslash in(0,n).$

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

:: $n>9\; \backslash left(\backslash frac\{1-p\}\{p\}\; \backslash right)\backslash quad\backslash text\{and\}\backslash quad\; n>9\backslash left(\backslash frac\{p\}\{1-p\}\backslash right).$

{{hidden begin|style=width:66%|ta1=center|border=1px #aaa solid|title=[Proof]}}

The rule $np\backslash pm3\backslash sqrt\{np(1-p)\}\backslash in(0,n)$ is totally equivalent to request that

:

Moving terms around yields:

: $np>3\backslash sqrt\{np(1-p)\}\backslash quad\backslash text\{and\}\backslash quad\; n(1-p)>3\backslash sqrt\{np(1-p)\}.$

Since

: $n>9\; \backslash left(\backslash frac\{1-p\}p\backslash right)\; \backslash quad\backslash text\{and\}\backslash quad\; n>9\; \backslash left(\backslash frac\{p\}\{1-p\}\backslash right).$

Notice that these conditions automatically imply that $n>9$. On the other hand, apply again the square root and divide by 3,

: $\backslash frac\{\backslash sqrt\{n\}\}3>\backslash sqrt\{\backslash frac\{1-p\}p\}>0\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash frac\{\backslash sqrt\{n\}\}3\; >\; \backslash sqrt\{\backslash frac\{p\}\{1-p\}\}>0.$

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

: $\backslash frac\{\backslash sqrt\{n\}\}3>\backslash sqrt\{\backslash frac\{1-p\}p\}-\backslash sqrt\{\backslash frac\{p\}\{1-p\}\}>-\backslash frac\{\backslash sqrt\{n\}\}3;$

and so, the desired first rule is satisfied,

:

{{hidden end}}

• Another commonly used rule is that both values {{math|np}} and {{math|n(1 − p)}} must be greater than{{Cite book |last=Chen |first=Zac |title=H2 Mathematics Handbook |publisher=Educational Publishing House |year=2011 |isbn=9789814288484 |edition=1 |location=Singapore |pages=350}}{{Cite web |date=2023-05-29 |title=6.4: Normal Approximation to the Binomial Distribution - Statistics LibreTexts |url=https://stats.libretexts.org/Courses/Las_Positas_College/Math_40:_Statistics_and_Probability/06:_Continuous_Random_Variables_and_the_Normal_Distribution/6.04:_Normal_Approximation_to_the_Binomial_Distribution |access-date=2023-10-07 |archive-date=2023-05-29 |archive-url=https://web.archive.org/web/20230529211919/https://stats.libretexts.org/Courses/Las_Positas_College/Math_40:_Statistics_and_Probability/06:_Continuous_Random_Variables_and_the_Normal_Distribution/6.04:_Normal_Approximation_to_the_Binomial_Distribution |url-status=bot: unknown }} 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.

{{hidden begin|style=width:66%|ta1=center|border=1px #aaa solid|title=[Proof]}}

Assume that both values $np$ and $n(1-p)$ are greater than 9. Since , we easily have that

: $np\backslash geq9>9(1-p)\backslash quad\backslash text\{and\}\backslash quad\; n(1-p)\backslash geq9>9p.$

We only have to divide now by the respective factors $p$ and $1-p$, to deduce the alternative form of the 3-standard-deviation rule:

: $n>9\; \backslash left(\backslash frac\{1-p\}p\backslash right)\; \backslash quad\backslash text\{and\}\backslash quad\; n>9\; \backslash left(\backslash frac\{p\}\{1-p\}\backslash right).$

{{hidden end}}

The following is an example of applying a continuity correction. Suppose one wishes to calculate {{math|Pr(X ≤ 8)}} for a binomial random variable {{math|X}}. If {{math|Y}} has a distribution given by the normal approximation, then {{math|Pr(X ≤ 8)}} is approximated by {{math|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 {{math|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 1738. Nowadays, it can be seen as a consequence of the central limit theorem since {{math|B(n, p)}} is a sum of {{math|n}} independent, identically distributed Bernoulli variables with parameter {{math|p}}. This fact is the basis of a hypothesis test, a "proportion z-test", for the value of {{math|p}} using {{math|x/n}}, the sample proportion and estimator of {{math|p}}, in a common test statistic.NIST/SEMATECH, [http://www.itl.nist.gov/div898/handbook/prc/section2/prc24.htm "7.2.4. Does the proportion of defectives meet requirements?"] e-Handbook of Statistical Methods.

For example, suppose one randomly samples {{math|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

: $\backslash sigma\; =\; \backslash sqrt\{\backslash frac\{p(1-p)\}\{n\}\}$

The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product {{math|np}} converges to a finite limit. Therefore, the Poisson distribution with parameter {{math|1=λ = np}} can be used as an approximation to {{math|B(n, p)}} of the binomial distribution if {{math|n}} is sufficiently large and {{math|p}} is sufficiently small. According to rules of thumb, this approximation is good if {{math|n ≥ 20}} and {{math|p ≤ 0.05}}{{cite news |date=2023-03-28 |title=12.4 – Approximating the Binomial Distribution {{!}} STAT 414 |newspaper=Pennstate: Statistics Online Courses |url=https://online.stat.psu.edu/stat414/lesson/12/12.4 |access-date=2023-10-08 |archive-date=2023-03-28 |archive-url=https://web.archive.org/web/20230328081322/https://online.stat.psu.edu/stat414/lesson/12/12.4 |url-status=bot: unknown }} such that {{math|np ≤ 1}}, or if {{math|n > 50}} and {{math|p < 0.1}} such that {{math|np < 5}},{{Cite book |last=Chen |first=Zac |title=H2 mathematics handbook |publisher=Educational publishing house |year=2011 |isbn=9789814288484 |edition=1 |location=Singapore |pages=348}} or if {{math|n ≥ 100}} and {{math|np ≤ 10}}.NIST/SEMATECH, [http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm "6.3.3.1. Counts Control Charts"], e-Handbook of Statistical Methods.{{Cite web |date=2023-03-13 |title=The Connection Between the Poisson and Binomial Distributions |url=https://mathcenter.oxford.emory.edu/site/math117/connectingPoissonAndBinomial/ |access-date=2023-10-08 |archive-date=2023-03-13 |archive-url=https://web.archive.org/web/20230313085931/https://mathcenter.oxford.emory.edu/site/math117/connectingPoissonAndBinomial/ |url-status=bot: unknown }}

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. {{ISBN|9781-43983-5746}}. ch. 4, and references therein.

• Poisson limit theorem: As {{math|n}} approaches {{math|∞}} and {{math|p}} approaches 0 with the product {{math|np}} held fixed, the {{math|Binomial(n, p)}} distribution approaches the Poisson distribution with expected value {{math|1=λ = np}}.
• de Moivre–Laplace theorem: As {{math|n}} approaches {{math|∞}} while {{math|p}} remains fixed, the distribution of
• : $\backslash frac\{X-np\}\{\backslash sqrt\{np(1-p)\}\}$

: approaches the normal distribution with expected value 0 and variance 1. This result is sometimes loosely stated by saying that the distribution of {{math|X}} is asymptotically normal with expected value 0 and variance 1. This result is a specific case of the central limit theorem.

The binomial distribution and beta distribution are different views of the same model of repeated Bernoulli trials. The binomial distribution is the PMF of {{mvar|k}} successes given {{mvar|n}} independent events each with a probability {{mvar|p}} of success.

Mathematically, when {{math|1=α = k + 1}} and {{math|1=β = n − k + 1}}, 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 {{math|n + 1}}:

: $\backslash operatorname\{Beta\}(p;\backslash alpha;\backslash beta)\; =\; (n+1)B(k;n;p)$

Beta distributions also provide a family of prior probability distributions for binomial distributions in Bayesian inference:{{cite book| last=MacKay| first=David| title = Information Theory, Inference and Learning Algorithms|year=2003| publisher=Cambridge University Press; First Edition |isbn=978-0521642989}}

: $P(p;\backslash alpha,\backslash beta)\; =\; \backslash frac\{p^\{\backslash alpha-1\}(1-p)^\{\backslash beta-1\}\}\{\backslash operatorname\{Beta\}(\backslash alpha,\backslash beta)\}.$

Given a uniform prior, the posterior distribution for the probability of success {{mvar|p}} given {{mvar|n}} independent events with {{mvar|k}} observed successes is a beta distribution.{{Cite web|url=https://www.statlect.com/probability-distributions/beta-distribution|title = Beta distribution}}

{{further|Pseudo-random number sampling}}

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 [http://luc.devroye.org/chapter_ten.pdf Chapter X, Discrete Univariate Distributions])

{{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

}}

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 {{math|1=Pr(X = k)}} for all values {{mvar|k}} from {{math|0}} through {{mvar|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 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.

This distribution was derived by Jacob Bernoulli. He considered the case where {{math|1=p = r/(r + s)}} where {{math|p}} is the probability of success and {{math|r}} and {{math|s}} are positive integers. Blaise Pascal had earlier considered the case where {{math|1=p = 1/2}}, tabulating the corresponding binomial coefficients in what is now recognized as Pascal's triangle.{{cite book|last=Katz|first=Victor|title=A History of Mathematics: An Introduction|publisher=Addison-Wesley|year=2009|isbn=978-0-321-38700-4|pages=491|chapter=14.3: Elementary Probability}}

{{Portal|Mathematics}}

• 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

{{reflist|colwidth=30em}}

{{reflist|group=note}}

• {{cite book |first=Werner Z. |last=Hirsch |title=Introduction to Modern Statistics |location=New York |publisher=MacMillan |year=1957 |chapter=Binomial Distribution—Success or Failure, How Likely Are They? |pages=140–153 |chapter-url=https://books.google.com/books?id=KostAAAAIAAJ&pg=PA140 }}
• {{cite book |first1=John |last1=Neter |first2=William |last2=Wasserman |first3=G. A. |last3=Whitmore |title=Applied Statistics |location=Boston |publisher=Allyn & Bacon |edition=Third |year=1988 |isbn=0-205-10328-6 |pages=185–192 }}

{{Commons category|Binomial distributions}}{{Wikifunctions|Z20094|}}

• Interactive graphic: [http://www.math.wm.edu/~leemis/chart/UDR/UDR.html Univariate Distribution Relationships]
• [http://www.fxsolver.com/browse/formulas/Binomial+distribution Binomial distribution formula calculator]
• Difference of two binomial variables: [https://math.stackexchange.com/q/1065487 X-Y] or [https://math.stackexchange.com/q/562119 |X-Y|]
• [http://www.wolframalpha.com/input/?i=Prob+x+%3E+19+if+x+is+binomial+with+n+%3D+36++and+p+%3D+.6 Querying the binomial probability distribution in WolframAlpha]
• Confidence (credible) intervals for binomial probability, p: [https://causascientia.org/math_stat/ProportionCI.html online calculator] available at [https://causascientia.org causaScientia.org]

{{-}}

{{ProbDistributions|discrete-finite}}

{{DEFAULTSORT:Binomial Distribution}}

Category:Discrete distributions

Category:Factorial and binomial topics

Category:Conjugate prior distributions

Category:Exponential family distributions