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| The '''Bernoulli distribution''' is a discrete probability distribution that gives 1 with probability ''p'' and 0 with probability ''q = 1 - p''. | The '''Bernoulli distribution''' is a discrete probability density function, specifically giving outcomes 0 or 1. |
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| The distribution is appropriate for modeling any binary outcome. | The distribution gives outcome 1 with probability ''p'', and 0 with probability ''q = 1 - p''. It is appropriate for modeling any binary event. |
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| The sum of repeated and independent Bernoulli-distributed events are described by the [[Statistics/BinomialDistribution|binomial distribution]]. | 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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| == Statistics == | == Moments == |
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| The expected value is ''E[X] = p''. | The [[Statistics/Moments|first moment]] is ''E[X] = p''. |
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| The variance is ''Var[X] = p(1 - p) = pq''. | The [[Statistics/Variance|variance]] is ''Var[X] = p(1 - p) = pq''. |
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| The expected number of cases sampled is ''np''; the sample size is described by the [[Statistics/BinomialDistribution|binomial distribution]]. | The expected number of cases sampled is ''np''; the sample size is described by the [[Analysis/BinomialDistribution|binomial distribution]]. |
Bernoulli Distribution
The Bernoulli distribution is a discrete probability density function, specifically giving outcomes 0 or 1.
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 binomial distribution.
Moments
The first moment is E[X] = p.
The variance is Var[X] = p(1 - p) = pq.
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 binomial distribution.