View source for Binomial distribution

[[Category:Discrete distributions]]
[[Category:Bounded distributions]]
[[Category:Unimodal distributions]]
[[Category:Univariate distributions]]
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Consider an event—such as a coin coming down heads—that can be true or false in each trial—or each toss—with probability «p» -- it has a [[Bernoulli]] distribution. A binomial distribution describes the number of times an event is true -- e.g., the coin is heads -- in «n» independent trials—or tosses—where the event occurs with probability «p» on each trial.

<center>[[image:BinomialDistribution.png]]</center>

== Functions ==
=== Parameters ===
The binomial distribution is described by
* «n»: The number of trials
* «p»: The probability of success in each trial
* «k»: Number of successes
* «u» : The overall probability of ''k'' successes in «n» trials (where ''k'' is the result).  This is also known as the fractile or quartile level of the outcome. Must be ''0 &le; «u» &le; 1''.

=== Binomial(n, p) ===
The number of successes for a quantity described by a binomial distribution. This is the Poisson [[:category|Distribution function]]. Use this to describe an uncertain variable that represents the number of successes in a repeated trial.

=== <div id="ProbBinomial">Prob{{Release||5.1|_}}Binomial(k, n, p)</div> ===
{{Release||5.1|To use, add the [[Distribution Densities Library]] to your model.}}

The Binomial distribution is a non-negative discrete distribution where the probability of outcome ''k'' is given by 
:<math>P_{n,p}(k) = \left(\begin{array}{c}n\\k\end{array}\right) p^k (1-p)^{n-k}</math>

=== <div id="CumBinomial">CumBinomial(k, n, p)</div> ===

The [[:category:Cumulative distribution functions|analytic cumulative probability function]] for a Poisson distribution. Returns the probability that the actual number of successes is less than or equal to «k».

=== <div id="CumBinomialInv">CumBinomialInv(u, n, p)</div> ===

The [[:category:Inverse cumulative probability functions|analytic inverse cumulative probability function]] for a Poisson(n,p) distribution, also known as the ''quantile function''. For 0<=u<1, this returns the number of successes, k, so that the probability of the outcome of «n» trials having «k» or fewer successes is less than or equal to «k».

==== Examples ====
An experiment is to be conducted that will consist of rolling a die 100 times and counting the number of times a 6 is rolled.  With a 75% probability, the number of sixes that will be observe will be less than or equal to:
:<code>CumBinomialInv(75%, 100, 1/6 )</code>

[[CumBinomialInv]] is the inverse of [[CumBinomial]], so that for all ''«n» &ge; 0, 0 &le; k&le; «n»'', and ''0 < «p» < 1'', the following holds (up to the available numeric precision):
:<code>CumBinomialInv(CumBinomial(k, n, p), n, p) &rarr; k</code>

== Statistics ==
The distribution has the following statistics:
* [[Mean]] = <math>n p</math> 
* [[Variance]] = <math>n p (1 - p)</math>
* [[Skewness]] = <math>{{1-2 p}\over{\sqrt{n p (1-p)}}}</math>
* [[Kurtosis]] = <math>{{1 - 6 p (1-p)}\over{n p (1-p)}}</math>

== See Also ==
* [[Multinomial]] -- A generalization of Binomial in which more than two outcomes are possible.
* [[NegativeBinomial]] -- the two other common discrete distributions on the non-negative integers
* [[Poisson distribution]]
* [[Normal distribution]]
* [[Parametric discrete distributions]]
* [[Distribution Densities Library]]