Solved: Let X1 and X2 be independent chi-square random

Chapter 5, Problem 7E

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QUESTION:

Let \(X_{1}\) and \(X_{2}\) be independent chi-square random variables with \(r_{1}\) and \(r_{2}\) degrees of freedom, respectively. Show that

(a) \(U=X_{1} /\left(X_{1}+X_{2}\right)\) has a beta distribution with \(\alpha=r_{1} / 2\) and \(\beta=r_{2} / 2\).

(b) \(V=X_{2} /\left(X_{1}+X_{2}\right)\) has a beta distribution with \(\alpha=r_{2} / 2\) and \(\beta=r_{1} / 2\).

Equation Transcription:

 

 

 

 


 

 


 

 


 

Text Transcription:

X_1  

X_2  

r_1  

r_2  

U=X_1/X_1+X_2  

alpha=r_1/2  

beta=r_2/2  

V=X_2/(X_1+X_2)  

alpha=r_2/2  

beta=r_1/2  

Questions & Answers

QUESTION:

Let \(X_{1}\) and \(X_{2}\) be independent chi-square random variables with \(r_{1}\) and \(r_{2}\) degrees of freedom, respectively. Show that

(a) \(U=X_{1} /\left(X_{1}+X_{2}\right)\) has a beta distribution with \(\alpha=r_{1} / 2\) and \(\beta=r_{2} / 2\).

(b) \(V=X_{2} /\left(X_{1}+X_{2}\right)\) has a beta distribution with \(\alpha=r_{2} / 2\) and \(\beta=r_{1} / 2\).

Equation Transcription:

 

 

 

 


 

 


 

 


 

Text Transcription:

X_1  

X_2  

r_1  

r_2  

U=X_1/X_1+X_2  

alpha=r_1/2  

beta=r_2/2  

V=X_2/(X_1+X_2)  

alpha=r_2/2  

beta=r_1/2  

ANSWER:

Step 1 of 6

Given:

  and  are independent chi-square random variables with   and degrees of freedom, respectively.

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