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| ## page was renamed from Statistics/BernoulliDistribution | |
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| The sum of repeated and independent Bernoulli-distributed events are described by the [[Statistics/BinomialDistribution|binomial distribution]]. | The sum of repeated and independent Bernoulli-distributed events are described by the [[Analysis/BinomialDistribution|binomial distribution]]. |
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| The [[Statistics/Moments|first moment]] is ''E[X] = p''. | The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = p''. |
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| The [[Statistics/Variance|variance]] is ''Var[X] = p(1 - p) = pq''. | [[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}} Furthermore note that ''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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| === 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)''. |
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| If all frame listings have an equal probability of selection, sampling can be implemented like: | Selection for an experiment can be conceptualized as a Bernoulli event. The natural implementation of a sample is: |
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| The expected number of cases sampled is ''np''; the sample size is described by the [[Statistics/BinomialDistribution|binomial distribution]]. | 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}} |
Bernoulli Distribution
The Bernoulli distribution is a discrete probability density function, specifically giving outcomes 0 or 1.
Contents
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 expected value is given as E[X] = p.
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:
Furthermore note that E[X] = p, that Prob(X=1) = p, and that Prob(X=0) = 1 - p:
It should be clear that this simplifies to p(1-p) immediately.
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 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 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 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 normal as nX̅ ~ N(X̅, X̅(1-X̅)/n).
This leads to a simple formula for the Wald interval:
