Given T~ Gam(2, 0) with shape 2 and scale 0 >0 - measured in years. Let Z = XT for some constant > > 0. In addition, given Y > 0, which follows a Y~ Gam(2, n) with shape 2 and scale n >0 and PQ is defined as PQ = Z - Y. Assuming Y Z and that random variable X, with the Gam(a, scale = B) density is 1 Για) βα xa-1e-x/B a, ß >0 f(x, a, B) for x >0 and f(x; a, B) = 0 elsewhere. a. Find the density of the transformation Z = XT, fz(z). b. What is the joint density fzy (z, y)? Show that it is a valid density. =
Given T~ Gam(2, 0) with shape 2 and scale 0 >0 - measured in years. Let Z = XT for some constant > > 0. In addition, given Y > 0, which follows a Y~ Gam(2, n) with shape 2 and scale n >0 and PQ is defined as PQ = Z - Y. Assuming Y Z and that random variable X, with the Gam(a, scale = B) density is 1 Για) βα xa-1e-x/B a, ß >0 f(x, a, B) for x >0 and f(x; a, B) = 0 elsewhere. a. Find the density of the transformation Z = XT, fz(z). b. What is the joint density fzy (z, y)? Show that it is a valid density. =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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