= Bernoulli Distribution =

The '''Bernoulli distribution''' is a discrete probability density function, specifically giving outcomes 0 or 1.

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== Description ==

The distribution gives outcome 1 with probability ''p'', and 0 with probability ''q = 1 - p''. It is appropriate for modeling any binary event.

A variable distributed this way is notated like ''X ~ Bernoulli(p)''. (Sometimes shortened to 'Bern'.)

The sum of repeated and independent Bernoulli-distributed events are described by the [[Analysis/BinomialDistribution|binomial distribution]].

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== Moments ==

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

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

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== Usage ==



=== Sampling ===

If all frame listings have an equal probability of selection, sampling can be implemented like:

{{{
scalar p = .2 /* Probability of selection */
set seed 123456789
generate double r = runiform()
generate sampled = (r < p)
}}}

The expected number of cases sampled is ''np''; the sample size is described by the [[Analysis/BinomialDistribution|binomial distribution]].



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