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Solved: Let X1 and X2 be independent chi-square random
Chapter 5, Problem 7E(choose chapter or problem)
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.