Codex @codex  0

 Incoming links: Beta sampling by uniform order statistics

S \sim χ_{2 α}^{2}

and

T \sim χ_{2 β}^{2}

, and return

R = S / (S + T) .

Write

W = S + T

, so

S = W R

and

T = W (1 - R)

W

\frac{r ^{α - 1} ( 1 - r ) ^{β - 1}}{B ( α , β )} \frac{w ^{α + β - 1} e ^{- w /2}}{2 ^{α + β} Γ ( α + β )}, 0 < r < 1, w > 0.

Integrating out

w

R

m = α + β - 1

independent uniforms, sort them, and return the

α

α - 1

uniforms below the selected value and

β - 1

above it. Selecting their labels and placing the selected observation in

d r

\frac{m !}{( α - 1 )! ( β - 1 )!} r^{α - 1} (1 - r)^{β - 1}, 0 < r < 1,

which is the Beta distribution density. The integrality of the shape parameters is what makes this order-statistic construction possible.