= Binomial Distribution =

The '''binomial distribution''' is a discrete probability density function.

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== Description ==

The distribution describes the sum of repeated and independent [[Analysis/BernoulliDistribution|Bernoulli-distributed]] events.

The probability of ''k'' successes can be thought of as the number of ways ''k'' can be selected from ''n'' Bernoulli trials. Naturally the density function is parameterized in terms of ''n'', ''k'', and ''p''.

{{attachment:binom.svg}}

for integer values of ''k ∈ [0,n]''.

It follows that the cumulative density function is:

{{attachment:cumbinom.svg}}

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== Moments ==

The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = np''.

[[Analysis/Variance|Variance]] is given as ''Var[X] = np(1 - p) = npq''.

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== Usage ==



=== Sum of Repeated Bernoulli Trials ===

The sum of repeated (independent) [[Analysis/BernoulliDistribution|Bernoulli trials]] is known to follow a binomial distribution. In other words, if ''X ~ Bernoulli(p)'' then ''nX̅ ~ Binomial(n,p)''. 



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CategoryRicottone