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Refer to Exercise 6.77. If Y1, Y2, . . . , Yn are
Chapter 6, Problem 79E(choose chapter or problem)
Refer to Exercise 6.77. If \(Y_1,\ Y_2,\ldots,\ Y_n\) are independent, uniformly distributed random variables on the interval \([0,\ \theta]\), show that \(U=Y_{(1)} / Y_{(n)}\) and \(Y_{(n)}\) are independent.
Questions & Answers
QUESTION:
Refer to Exercise 6.77. If \(Y_1,\ Y_2,\ldots,\ Y_n\) are independent, uniformly distributed random variables on the interval \([0,\ \theta]\), show that \(U=Y_{(1)} / Y_{(n)}\) and \(Y_{(n)}\) are independent.
ANSWER:Refer to Exercise 6.77. If Y1, Y2, . . . , Yn are independent, uniformly distributedrandom variables on the interval [0, ], show that U = Y(1) /Y( n) nd Y(n) areindependent.ReferenceLet Y1, , . . . , Yn be independent, uniformly distributed random variables on theinterval [0, ].a Find the joint density function of Y( j ) and Y(k) where j and k are integers 1 j < k n.b Use the result from part (a) to find Cov(Y( j ) , Y(k) ) hen j and k are integers 1 j < k n.c .Use the result from part (b)