Beta distribution

A beta distributed random variable $𝑋 \sim B e t a (𝑎, 𝑏)$ is described by the following probability density function #m/def/prob

𝑓 𝑋 (𝑥) = 1 𝛽 (𝑎, 𝑏) 𝑥 𝑎 - 1 (1 - 𝑥) 𝑏 - 1

where $𝛽 (𝑎, 𝑏)$ is chosen so as to normalize $𝑓 𝑋$

𝛽 (𝑎, 𝑏) = Γ (𝑎 + 𝑏) Γ (𝑎) Γ (𝑏)

Note $B e t a (1, 1) \sim U n i f (0, 1)$ .

Properties

Furthermore

Let $𝑋 𝑖 \sim U (0, 1)$ be independently distributed. Then the $𝑗$ th Order statistic $𝑋 (𝑗) \sim B e t a (𝑗, 𝑛 - 𝑗 + 1)$ .

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