Continuous random variable

Beta distribution

A beta distributed random variable 𝑋 ∼Beta(𝑎,𝑏) is described by the following probability density function prob

𝑓𝑋(𝑥)=1𝛽(𝑎,𝑏)𝑥𝑎−1(1−𝑥)𝑏−1

where 𝛽(𝑎,𝑏) is chosen so as to normalize 𝑓𝑋

𝛽(𝑎,𝑏)=Γ(𝑎+𝑏)Γ(𝑎)Γ(𝑏)

Note Beta(1,1) ∼Unif(0,1).

Properties

Furthermore

1. Let 𝑋𝑖 ∼U(0,1) be independently distributed. Then the 𝑗th Order statistic 𝑋(𝑗) ∼Beta(𝑗,𝑛 −𝑗 +1).

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