= Bernoulli Distribution =

The '''Bernoulli distribution''' is a discrete probability mass 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.

Or more formally:
 * ''Prob(X=1) = p''
 * ''Prob(X=0) = 1 - p''

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''. Observe that the standard calculation for variance of a discrete quantitative random variable simplifies quickly, since there are only two possible values:

{{attachment:var1.svg}}

Make the substitutions for ''E[X] = p'', that ''Prob(X=1) = p'', and that ''Prob(X=0) = (1 - p)'':

{{attachment:var2.svg}}

It should be clear that this simplifies to ''p(1-p)'' immediately.

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== Usage ==



=== Bernoulli Trials ===

Consider a Bernoulli random variable with probability ''p''. A single trial to observe this variable is called a '''Bernoulli trial'''. The sum of repeated (independent) Bernoulli trials is known to follow a [[Analysis/BinomialDistribution|binomial distribution]]. In other words, if ''X ~ Bernoulli(p)'' then ''nX̅ ~ Binomial(n,p)''. 



=== Sampling ===

Selection for an experiment can be conceptualized as a Bernoulli event.

The natural implementation of a sample is:

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

The sample size ''np'' is known to follow a [[Analysis/BinomialDistribution|binomial distribution]].



=== Conservative Confidence Interval ===

The variance of a Bernoulli random variable, i.e. ''p(1 - p)'', is maximized by ''p = 1/2''. (The same is true for a [[Analysis/BinomialDistribution|binomial]] random variable; ''Var[X] = np(1 - p)''.) Therefore ''Var[X] ≤ 1/4''. This leads to the concept of a '''conservative confidence interval''' that simply assumes the 'worst case' and sets variance to ''1/4''.

Following from the '''De Moivre-Laplace theorem''', with a large enough ''n'', a binomial random variable is approximately [[Analysis/NormalDistribution|normal]] as ''nX̅ ~ N(X̅, X̅(1-X̅)/n)''.

This leads to a simple formula for the '''Wald interval''':

{{attachment:wald.svg}}



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