BinomialDistribution

Generated from the engine's own definition: compute-engine's symbol, which we neither extend nor document by hand.

Binomial distribution: number of successes in n independent trials, each with success probability p.

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BinomialDistribution(integer<0..>, real<0..1>) -> expression<BinomialDistribution>as compute-engine declares it

Typed signatures

Type	Package	Applies
(integer<0..>, real<0..1>) -> expression<BinomialDistribution>	compute-engine	always

Domain: Compute engine

Examples

Basic #

The mean is $np$

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In$\mathrm{mean}(\operatorname{BinomialDistribution}(n, p))$Out$np$

The variance is $np(1-p)$

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In$\mathrm{Variance}(\operatorname{BinomialDistribution}(n, p))$Out$np×(1-p)$

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In$\operatorname{PDF}(\operatorname{BinomialDistribution}(n, p), k)$Out$\begin{cases}\binom{n}{k}p^{k}(1-p)^{n-k}&\lfloor k\rfloor=k∧0≤k∧k≤n\\0&\mathrm{True}\end{cases}$

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In$\operatorname{CDF}(\operatorname{BinomialDistribution}(n, p), k)$Out$\begin{cases}0&k\lt0\\1&n≤k\\\operatorname{BetaRegularized}(1-p, n-\lfloor k\rfloor, \lfloor k\rfloor+1)&\mathrm{True}\end{cases}$

Properties #

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In$\operatorname{PDF}(\operatorname{BinomialDistribution}(n, p), x)$Out$\begin{cases}\binom{n}{x}p^{x}(1-p)^{n-x}&\lfloor x\rfloor=x∧0≤x∧x≤n\\0&\mathrm{True}\end{cases}$

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In$\mathrm{FunctionExpand}(\operatorname{PDF}(\operatorname{BinomialDistribution}(n, p), k))$Out$\begin{cases}\binom{n}{k}p^{k}(1-p)^{n-k}&\lfloor k\rfloor=k∧0≤k∧k≤n\\0&\mathrm{True}\end{cases}$

About

Examples	6
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