9F (Probability)

U = X / (X + Y)

and

V = X + Y

. Their inverse change of variables is

X = U V

Y = (1 - U) V

, with domain

0 < U < 1

V > 0

. The absolute Jacobian determinant of this inverse transformation is

det (v - v u 1 - u) = v .

f_{U, V} (u, v) = \frac{θ ^{n + m}}{( n - 1 )! ( m - 1 )!} u^{n - 1} (1 - u)^{m - 1} v^{n + m - 1} e^{- θ v}, 0 < u < 1, v > 0.

Factor this into two normalized densities:

f_{U, V} (u, v) = [\frac{( n + m - 1 )!}{( n - 1 )! ( m - 1 )!} u^{n - 1} (1 - u)^{m - 1}] [\frac{θ ^{n + m}}{( n + m - 1 )!} v^{n + m - 1} e^{- θ v}] .

The support is a product domain, and the factors are precisely a Beta distribution density and a Gamma distribution density. Therefore

U \sim B (n, m), V \sim Γ (n + m, θ), U and V independent .

θ

; it is a rate parameter, not the scale parameter

1/ θ

 Ancestors (9)

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