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Statistics[Distributions]

 Binomial
 binomial distribution

 Calling Sequence Binomial(n, p) BinomialDistribution(n, p)

Parameters

 n - number of trials p - probability of success

Description

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

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

 subject to the following conditions:

$0

 • The binomial distribution is used for the estimation of probabilities in a set of success or failures.  The binomial variate indicates the number of successes in a set of n Bernoulli trials, each with probability of success p.
 • Note that the Binomial command is inert and should be used in combination with the RandomVariable command.

Notes

 • The Quantile function applied to a binomial distribution uses a sequence of iterations in order to converge on the desired output point.  The maximum number of iterations to perform is equal to 100 by default, but this value can be changed by setting the environment variable _EnvStatisticsIterations to the desired number of iterations.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Binomial}\left(n,p\right)\right):$
 > $\mathrm{ProbabilityFunction}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \left(\genfrac{}{}{0}{}{{n}}{{u}}\right){}{{p}}^{{u}}{}{\left({1}{-}{p}\right)}^{{n}{-}{u}}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{ProbabilityFunction}\left(X,1\right)$
 ${n}{}{p}{}{\left({1}{-}{p}\right)}^{{n}{-}{1}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${p}{}{n}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${n}{}{p}{}\left({1}{-}{p}\right)$ (4)

References

 Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.
 Johnson, Norman L.; 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.