Paper 2, Section II, F

Probability | Part IA, 2016

For any positive integer nn and positive real number θ\theta, the Gamma distribution Γ(n,θ)\Gamma(n, \theta) has density fΓf_{\Gamma} defined on (0,)(0, \infty) by

fΓ(x)=θn(n1)!xn1eθx.f_{\Gamma}(x)=\frac{\theta^{n}}{(n-1) !} x^{n-1} e^{-\theta x} .

For any positive integers aa and bb, the Beta distribution B(a,b)B(a, b) has density fBf_{B} defined on (0,1)(0,1) by

fB(x)=(a+b1)!(a1)!(b1)!xa1(1x)b1f_{B}(x)=\frac{(a+b-1) !}{(a-1) !(b-1) !} x^{a-1}(1-x)^{b-1}

Let XX and YY be independent random variables with respective distributions Γ(n,θ)\Gamma(n, \theta) and Γ(m,θ)\Gamma(m, \theta). Show that the random variables X/(X+Y)X /(X+Y) and X+YX+Y are independent and give their distributions.

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