View source for Binomial distribution

{{Probability distribution|
  name       =Binomial|
  type       =mass|
  pdf_image  =[[Image:Binomial distribution pmf.png|325px|Probability mass function for the binomial distribution]]<br /><small>The lines connecting the dots are added for clarity</small>|
  cdf_image  =[[Image:Binomial distribution cdf.png|325px|Cumulative distribution function for the binomial distribution]]<br /><small>Colors match the image above</small>|
  parameters =<math>n \geq 0</math> number of trials ([[integer]])<br /><math>0\leq p \leq 1</math> success probability ([[real number|real]])|
  support    =<math>k \in \{0,\dots,n\}\!</math>|
  pdf        =<math>{n\choose k} p^k (1-p)^{n-k} \!</math>|
  cdf        =<math>I_{1-p}(n-\lfloor k\rfloor, 1+\lfloor k\rfloor) \!</math>|
  mean       =<math>np\!</math>|
  median     =one of <math>\{\lfloor np\rfloor-1, \lfloor np\rfloor, \lfloor np\rfloor+1\}</math>|
  mode       =<math>\lfloor (n+1)\,p\rfloor\!</math>|
  variance   =<math>np(1-p)\!</math>|
  skewness   =<math>\frac{1-2p}{\sqrt{np(1-p)}}\!</math>|
  kurtosis   =<math>\frac{1-6p(1-p)}{np(1-p)}\!</math>|
  entropy    =<math> \frac{1}{2} \ln \left( 2 \pi n e p (1-p) \right) + O \left( \frac{1}{n} \right) </math>|
  mgf        =<math>(1-p + pe^t)^n \!</math>|
  char       =<math>(1-p + pe^{it})^n \!</math>|
}}

{{CMG}}
__NOTOC__


==Overview==
In [[probability theory]] and [[statistics]], the '''binomial distribution''' is the [[discrete probability distribution]] of the number of successes in a sequence of ''n'' [[statistical independence|independent]] yes/no experiments, each of which yields success with [[probability]] ''p''. Such a success/failure experiment is also called a Bernoulli experiment or [[Bernoulli trial]].  In fact, when ''n'' = 1, the binomial distribution ''is'' a [[Bernoulli distribution]].  The binomial distribution is the basis for the popular [[binomial test]] of [[statistical significance]].

==Examples==
An elementary example is this: Roll a standard die ten times and count the number of sixes. The distribution of this random number is a binomial distribution with ''n'' = 10 and ''p'' = 1/6.

As another example, assume 5% of a very large population to be green-eyed. You pick 100 people randomly. The number of green-eyed people you pick is a [[random variable]] ''X'' which follows a binomial distribution with ''n'' = 100 and ''p'' = 0.05.

==Specification==

===Probability mass function===

In general, if the random variable ''K'' follows the binomial distribution with parameters ''n'' and ''p'', we write ''K'' ~ B(''n'', ''p''). The probability of getting exactly ''k'' successes is given by the [[probability mass function]]:

:<math>f(k;n,p)={n\choose k}p^k(1-p)^{n-k}</math>

for ''k'' = 0, 1, 2, ..., ''n'' and where

:<math>{n\choose k}=\frac{n!}{k!(n-k)!}</math>

is the [[binomial coefficient]] (hence the name of the distribution) "''n'' choose ''k''" (also denoted ''C''(''n'', ''k'') or ''n''C''k''). The formula can be understood as follows: we want ''k'' successes (''p''<sup>''k''</sup>) and ''n'' &minus; ''k'' failures (1 &minus; ''p'')<sup>''n'' &minus; ''k''</sup>. However, the ''k'' successes can occur anywhere among the ''n'' trials, and there are C(''n'', ''k'') different ways of distributing ''k'' successes in a sequence of ''n'' trials.

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

:<math>f(k;n,p)=f(n-k;n,1-p).\,\!</math>

So, one must look to a different ''k'' and a different ''p'' (the binomial is not symmetrical in general).

===Cumulative distribution function===

The [[cumulative distribution function]] can be expressed in terms of the regularized incomplete beta function, as follows:

:<math> F(k;n,p) = \Pr(X \le k) = I_{1-p}(n-k, k+1) \!</math>

provided ''k'' is an integer and 0&nbsp;&le;&nbsp;''k''&nbsp;&le;&nbsp;''n''.  If ''x'' is not necessarily an integer or not necessarily positive, one can express it thus:

:<math>F(x;n,p) = \Pr(X \le x) = \sum_{j=0}^{\operatorname{Floor}(x)} {n\choose j}p^j(1-p)^{n-j}</math>

For ''k'' &le; ''np'', [[Chernoff bound|upper bounds]] for the lower tail of the distribution function can be derived.  In particular, [[Hoeffding's inequality]] yields the bound

:<math> F(k;n,p) \leq \exp\left(-2 \frac{(np-k)^2}{n}\right), \!</math>

and [[Chernoff's inequality]] can be used to derive the bound

:<math> F(k;n,p) \leq \exp\left(-\frac{1}{2\,p} \frac{(np-k)^2}{n}\right). \!</math>

==Mean, variance, and mode==

If ''X'' ~ B(''n'', ''p'') (that is, ''X'' is a binomially distributed random variable), then the [[expected value]] of ''X'' is

:<math>\operatorname{E}(X)=np\,\!</math>

and the [[variance]] is

:<math>\operatorname{Var}(X)=np(1-p).\,\!</math>  

This fact is easily proven as follows.  Suppose first that we have exactly one Bernoulli trial.  We have two possible outcomes, 1 and 0, with the first having probability ''p'' and the second having probability 1&nbsp;&minus;&nbsp;''p''; the mean for this trial is given by &mu;&nbsp;=&nbsp;''p''.  Using the definition of [[variance]], we have

:<math>\sigma^2= \left(1 - p\right)^2p + (0-p)^2(1 - p) = p(1-p).</math>

Now suppose that we want the variance for ''n'' such trials (i.e. for the general binomial distribution).  Since the trials are independent, we may add the variances for each trial, giving

:<math>\sigma^2_n = \sum_{k=1}^n \sigma^2 = np(1 - p). \quad</math>

The [[mode (statistics)|mode]] of ''X'' is the greatest integer less than or equal to (''n''&nbsp;+&nbsp;1)''p''; if ''m'' = (''n''&nbsp;+&nbsp;1)''p'' is an integer, then ''m''&nbsp;&minus;&nbsp;1 and ''m'' are both modes.

==Explicit derivations of mean and variance==

We derive these quantities from first principles. Certain particular sums occur in these two derivations.
We rearrange the sums and terms so that sums solely over complete binomial probability mass functions
([[Binomial Distribution#Probability mass function|pmf]]) arise, which are always unity

:<math> \sum_{k=0}^n \operatorname{Pr}(X=k) = \sum_{k=0}^n {n\choose k}p^k(1-p)^{n-k} = 1</math>

===Mean===

We apply the definition of the [[expected value]] of a [[discrete random variable]] to the binomial distribution

:<math>\operatorname{E}(X) = \sum_k x_k \cdot \operatorname{Pr}(x_k) = \sum_{k=0}^n k \cdot \operatorname{Pr}(X=k)

= \sum_{k=0}^n k \cdot {n\choose k}p^k(1-p)^{n-k}</math>

The first term of the series (with index ''k'' = 0) has value 0 since the first factor, ''k'', is zero.
It may thus be discarded, i.e. we can change the lower limit to: ''k'' = 1

:<math>\operatorname{E}(X) = \sum_{k=1}^n k \cdot \frac{n!}{k!(n-k)!} p^k(1-p)^{n-k}

=  \sum_{k=1}^n k \cdot \frac{n\cdot(n-1)!}{k\cdot(k-1)!(n-k)!} \cdot p \cdot p^{k-1}(1-p)^{n-k}</math>

We've pulled factors of ''n'' and ''k'' out of the factorials, and one power of ''p'' has been split off. We are preparing to redefine the indices.

:<math>\operatorname{E}(X) = np \cdot \sum_{k=1}^n \frac{(n-1)!}{(k-1)!(n-k)!} p^{k-1}(1-p)^{n-k}</math>

We rename ''m'' = ''n'' - 1 and ''s'' = ''k'' - 1. The value of the sum is not changed by this, but it now becomes readily recognizable

:<math>\operatorname{E}(X) = np \cdot \sum_{s=0}^m \frac{(m)!}{(s)!(m-s)!} p^s(1-p)^{m-s}
= np \cdot \sum_{s=0}^m {m\choose s} p^s(1-p)^{m-s}</math>

The ensuing sum is a sum over a complete binomial [[Binomial Distribution#Probability mass function|pmf]] (of one order lower than the initial sum, as it happens). Thus

:<math>\operatorname{E}(X) = np \cdot 1 = np</math>

===Variance===

It can be shown that the variance is equal to
(see: [[variance#Properties, formal|variance, 10. Computational formula for variance]]):

:<math>\operatorname{Var}(X) = \operatorname{E}(X^2) - (\operatorname{E}(X))^2.</math>

In using this formula we see that we now also need the expected value of ''X''<sup>2</sup>, which is

:<math>\operatorname{E}(X^2) = \sum_{k=0}^n k^2 \cdot \operatorname{Pr}(X=k)

= \sum_{k=0}^n k^2 \cdot {n\choose k}p^k(1-p)^{n-k}.</math>

We can use our experience gained above in deriving the mean. We know how to process one factor of ''k''. This gets us as far as

:<math>\operatorname{E}(X^2) = np \cdot \sum_{s=0}^m k \cdot {m\choose s} p^s(1-p)^{m-s}
= np \cdot \sum_{s=0}^m (s+1) \cdot {m\choose s} p^s(1-p)^{m-s}</math>

(again, with ''m'' = ''n'' - 1 and ''s'' = ''k'' - 1). We split the sum into two separate sums and we recognize each one

:<math>\operatorname{E}(X^2) = np \cdot \bigg( \sum_{s=0}^m s \cdot {m\choose s} p^s(1-p)^{m-s} + \sum_{s=0}^m 1 \cdot {m\choose s} p^s(1-p)^{m-s} \bigg).</math>

The first sum is identical in form to the one we calculated in the Mean (above). It sums to ''mp''.
The second sum is unity.

:<math>\operatorname{E}(X^2) = np \cdot ( mp + 1) = np((n-1)p + 1) = np(np - p + 1).</math>

Using this result in the expression for the variance, along with the Mean (E(''X'') = ''np''), we get

:<math>\operatorname{Var}(X) = \operatorname{E}(X^2) - (\operatorname{E}(X))^2 = np(np - p + 1) - (np)^2 = np(1-p).</math>

==Relationship to other distributions==

===Sums of binomials===
If ''X'' ~ B(''n'', ''p'') and ''Y'' ~ B(''m'', ''p'') are independent binomial variables, then ''X'' + ''Y''  is again a binomial variable; its distribution is 

:<math>X+Y \sim B(n+m, p).\,</math>

===Normal approximation===

[[image:BinDistApprox_large.png|right|250px|thumb|Binomial PDF and normal approximation for ''n'' = 6 and ''p'' = 0.5.]]

If ''n'' is large enough, the skew of the distribution is not too great, and a suitable [[continuity correction]] is used, then an excellent approximation to B(''n'', ''p'') is given by the [[normal distribution]]

:<math> \operatorname{N}(np, np(1-p)).\,\!</math>

Various [[Rule of thumb|rules of thumb]] may be used to decide whether ''n'' is large enough. One rule is that both ''np'' and ''n''(1 &minus; ''p'') must be greater than 5. However, the specific number varies from source to source, and depends on how good an approximation one wants; some sources give 10. Another commonly used rule holds that the above normal approximation is appropriate only if 

:<math>\mu \pm 3 \sigma = np \pm 3 \sqrt{np(1-p)} \in [0,n].</math>

The following is an example of applying a [[continuity correction]]: Suppose one wishes to calculate Pr(''X''&nbsp;&le;&nbsp;8) for a binomial random variable ''X''. If ''Y'' has a distribution given by the normal approximation, then Pr(''X''&nbsp;&le;&nbsp;8) is approximated by Pr(''Y''&nbsp;&le;&nbsp;8.5). The addition of 0.5 is the continuity correction; the uncorrected normal approximation gives considerably less accurate results.

This approximation is a huge time-saver (exact calculations with large ''n'' are very onerous); historically, it was the first use of the normal distribution, introduced in [[Abraham de Moivre]]'s book ''[[The Doctrine of Chances]]'' in 1733. Nowadays, it can be seen as a consequence of the central limit theorem since B(''n'', ''p'') is a sum of ''n'' independent, identically distributed 0-1 indicator variables.

For example, suppose you randomly sample ''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 you sampled groups of ''n'' people 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 &sigma; = (''p''(1 &minus; ''p'')''n'')<sup>1/2</sup>. Large [[sample size]]s ''n'' are good because the standard deviation gets smaller, which allows a more precise estimate of the unknown parameter ''p''.

===Poisson approximation===

The binomial distribution converges towards the [[Poisson distribution]] as the number of trials goes to infinity while the product ''np'' remains fixed. Therefore the Poisson distribution with parameter &lambda; = ''np'' can be used as an approximation to B(''n'', ''p'') of the binomial distribution if n is sufficiently large and p is sufficiently small.  According to two rules of thumb, this approximation is good if ''n'' &ge; 20 and ''p'' &le; 0.05, or if ''n'' &ge; 100 and ''np'' &le; 10.<ref>NIST/SEMATECH, '6.3.3.1. Counts Control Charts', ''e-Handbook of Statistical Methods'', <http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm> [accessed 25 October 2006]</ref>

==Limits of binomial distributions==

* As ''n'' approaches &infin; and ''p'' approaches 0 while ''np'' remains fixed at &lambda;&nbsp;>&nbsp;0 or at least ''np'' approaches &lambda;&nbsp;>&nbsp;0, then the Binomial(''n'',&nbsp;''p'') distribution approaches the [[Poisson distribution]] with [[expected value]] &lambda;.

* As ''n'' approaches &infin; while ''p'' remains fixed, the distribution of

::<math>{X-np \over \sqrt{np(1-p)\ }}</math>

:approaches the [[normal distribution]] with expected value 0 and [[variance]] 1 (this is just a specific case of the Central Limit Theorem).

==References==

<references/>

*{{cite paper | author=Abdi, H | title = [http://www.utdallas.edu/~herve/Abdi-Binomial2007-pretty.pdf] ((2007).  Binomial Distribution: Binomial and Sign Tests.. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.| year = 2007 |}}

* Luc Devroye, ''Non-Uniform Random Variate Generation'', New York: Springer-Verlag, 1986. ''See especially [http://cg.scs.carleton.ca/~luc/chapter_ten.pdf Chapter X, Discrete Univariate Distributions].''

* Voratas Kachitvichyanukul and Bruce W. Schmeiser, Binomial random variate generation, ''[[Communications of the ACM]]'' 31(2):216–222, February 1988. {{doi|10.1145/42372.42381}}

==See also==
*[[Bean machine]] / Galton board
*[[Beta distribution]]
*[[Hypergeometric distribution]]
*[[Multinomial distribution]]
*[[Negative binomial distribution]]
*[[Poisson distribution]]
*[[SOCR]]
*[[Normal distribution]]

==External links==

* [http://www.adsciengineering.com/bpdcalc/ Binomial Probability Distribution Calculator]
* [http://faculty.vassar.edu/lowry/binomialX.html Binomial Probabilities Simple Explanation]
* [http://www.socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Binomial Distribution Applet]
* [http://www.causeweb.org CAUSEweb.org] Many resources for teaching Statistics including Binomial Distribution

[[Category:Biostatistics]]
[[Category:Statistics]]

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