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Statistics[Distributions]

 Bernoulli
 Bernoulli distribution

 Calling Sequence Bernoulli(p) BernoulliDistribution(p)

Parameters

 p - probability of success

Description

 • The Bernoulli distribution is a discrete probability distribution with probability function given by:

$f\left(t\right)=\left\{\begin{array}{cc}1-p& t=0\\ p& t=1\\ 0& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0\le p,p\le 1$

 • The Bernoulli distribution comes about as a consequence of a single Bernoulli trial.  Success of the Bernoulli trial is indicated with x=1 and failure is indicated with x=0, where a success occurs with probability p.  The parameter p is also referred to as the Bernoulli probability parameter.
 • Note that the Bernoulli command is inert and should be used in combination with the RandomVariable command.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Bernoulli}\left(p\right)\right):$
 > $\mathrm{ProbabilityFunction}\left(X,0\right)$
 ${1}{-}{p}$ (1)
 > $\mathrm{ProbabilityFunction}\left(X,1\right)$
 ${p}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${p}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${p}{}\left({1}{-}{p}\right)$ (4)

References

 Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.
 Johnson, Norman L.; and Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.
 Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.