Exercise 6.46 If X and Y are independent random variables having the x2 distribution with m and ndegrees of freedom, respectively, prove that X+Y has the x2 distribution with m+n degrees of freedom.

From the given information, X and Y are independent variables with χ2 distribution with m and n degrees of freedom. Consider, the moment generating functions for X and Y, MX(t)=(1-2t)m2 MY(t)=(1-2t)n2Consider, the moment generating function for X+Y, MX+Y(t)=Ee(X+Y)t =EeXt×eYt =E(eXt)*E(eYt) since X and Y are independent variables =MX(t)*MY(t) =(1-2t)m2*(1-2t)n2 =(1-2t)m+n2 That is, the moment generating of X+Y is same the moment generating function of chi-square distribution with degrees of freedom m+n. Hence, the random variable X+Y has the χ2 distribution with m+n degrees of freedom.

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If X and Y are independent random variables having the x2 distribution with m and n dom, respectively, prove that X+Y has the x2 distribution with m+n degrees of freedom.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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