Editing Binomial distribution (section)

{{short description|Probability distribution}}
{{Redirect|Binomial model|the binomial model in options pricing|Binomial options pricing model}}
<!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion of standards used for probability distribution articles such as this one. -->
{{Probability distribution
  | name       = Binomial distribution
  | type       = mass
  | pdf_image  = [[File:Binomial distribution pmf.svg|300px|Probability mass function for the binomial distribution]]
  | cdf_image  = [[File:Binomial distribution cdf.svg|300px|Cumulative distribution function for the binomial distribution]]
  | notation   = <math>B(n,p)</math>
  | parameters = <math>n \in \{0, 1, 2, \ldots\}</math> &ndash; number of trials<br /><math>p \in [0,1]</math> &ndash; success probability for each trial<br /><math>q = 1 - p</math>
  | support    = <math>k \in \{0, 1, \ldots, n\}</math> &ndash; number of successes
  | pdf        = <math>\binom{n}{k} p^k q^{n-k}</math>
  | cdf        = <math>I_q(n - \lfloor k \rfloor, 1 + \lfloor k \rfloor)</math> (the [[Beta_function#Incomplete_beta_function|regularized incomplete beta function]])
  | mean       = <math>np</math>
  | median     = <math>\lfloor np \rfloor</math> or <math>\lceil np \rceil</math>
  | mode       = <math>\lfloor (n + 1)p \rfloor</math> or <math>\lceil (n + 1)p \rceil - 1</math>
  | variance   = <math>npq = np(1-p)</math>
  | skewness   = <math>\frac{q-p}{\sqrt{npq}}</math>
  | kurtosis   = <math>\frac{1-6pq}{npq}</math>
  | entropy    = <math>\frac{1}{2} \log_2 (2\pi enpq) + O \left( \frac{1}{n} \right)</math><br /> in [[Shannon (unit)|shannon]]s. For [[nat (unit)|nats]], use the natural log in the log.
  | mgf        = <math>(q + pe^t)^n</math>
  | char       = <math>(q + pe^{it})^n</math>
  | pgf        = <math>G(z) = [q + pz]^n</math>
  | fisher     = <math> g_n(p) = \frac{n}{pq} </math><br />(for fixed <math>n</math>)
}}
{{Probability fundamentals}}

[[File:Pascal's triangle; binomial distribution.svg|thumb|280px|Binomial distribution for {{math|p {{=}} 0.5}}<br />with {{mvar|n}} and {{mvar|k}} as in [[Pascal's triangle]]<br /><br />The probability that a ball in a [[Bean machine|Galton box]] with 8 layers ({{math|''n'' {{=}} 8}}) ends up in the central bin ({{math|''k'' {{=}} 4}}) is {{math|70/256}}.]]

In [[probability theory]] and [[statistics]], the '''binomial distribution''' with parameters {{mvar|n}} and {{mvar|p}} is the [[discrete probability distribution]] of the number of successes in a sequence of {{mvar|n}} [[statistical independence|independent]] [[experiment (probability theory)|experiment]]s, each asking a [[yes–no question]], and each with its own [[Boolean-valued function|Boolean]]-valued [[outcome (probability)|outcome]]: ''success'' (with probability {{mvar|p}}) or ''failure'' (with probability {{math|''q'' {{=}} 1 − ''p''}}). 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, i.e., {{math|''n'' {{=}} 1}}, the binomial distribution is a [[Bernoulli distribution]]. The binomial distribution is the basis for the [[binomial test]] of [[statistical significance]].<ref>{{Cite book|last=Westland|first=J. Christopher|title=Audit Analytics: Data Science for the Accounting Profession|publisher=Springer|year=2020|isbn=978-3-030-49091-1|location=Chicago, IL, USA|pages=53}}</ref>

The binomial distribution is frequently used to model the number of successes in a sample of size {{mvar|n}} drawn [[with replacement]] from a population of size {{mvar|N}}. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a [[hypergeometric distribution]], not a binomial one.  However, for {{mvar|N}} much larger than {{mvar|n}}, the binomial distribution remains a good approximation, and is widely used.