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

subject to the following conditions:

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.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Bernoulli}\left(p\right)\right)\:$

$\mathrm{ProbabilityFunction}\left(X\,0\right)$

$1-p$

$\mathrm{ProbabilityFunction}\left(X\,1\right)$

$p$

$\mathrm{Mean}\left(X\right)$

$p$

$\mathrm{Variance}\left(X\right)$

$p\left(1-p\right)$

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.