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Derive the joint density function for U1 and U2. Show that the marginal distribution of Uj is a beta distribution with parameters aj and a2. Show that the marginal distribution of Uz is a gamma distribution with parameters a = aj +a2 and ß. Establish that Uj and U2 are independent.

Derive the joint density function for U1 and U2. Show that the marginal distribution of Uj is a beta distribution with parameters aj and a2. Show that the marginal distribution of Uz is a gamma distribution with parameters a = aj +a2 and ß. Establish that Uj and U2 are independent.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Suppose that Y1 has a gamma distribution with parameters aj and ß, that Y¡ is gamma distributed with parameters az and ß, and that Y1 and Y2 are independent. Let U1 = Y¡/(Y1+ Y2) and U2= Y1+ Y2. a Derive the joint density function for U1 and U2. b Show that the marginal distribution of Uj is a beta distribution with parameters ɑj and a2. c Show that the marginal distribution of U2 is a gamma distribution with parameters a = aj +d2 and ß. d Establish that U1 and U2 are independent.
Transcribed Image Text:Suppose that Y1 has a gamma distribution with parameters aj and ß, that Y¡ is gamma distributed with parameters az and ß, and that Y1 and Y2 are independent. Let U1 = Y¡/(Y1+ Y2) and U2= Y1+ Y2. a Derive the joint density function for U1 and U2. b Show that the marginal distribution of Uj is a beta distribution with parameters ɑj and a2. c Show that the marginal distribution of U2 is a gamma distribution with parameters a = aj +d2 and ß. d Establish that U1 and U2 are independent.
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