Problem 79E

Refer to Exercise 6.77. If Y1, Y2, . . . , Yn are independent, uniformly distributed random variables on the interval [0, θ], show that U = Y(1) /Y(n) and Y(n) are independent.

Reference

Let Y1, Y2, . . . , Yn be independent, uniformly distributed random variables on the interval [0, θ].

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) ) when j and k are integers 1 ≤ j < k ≤ n.

c .Use the result from part (b) and Exercise 6.76 to find V (Y(k) − Y( j ) ), the variance of the difference between two order statistics.

Let Y1, Y2, . . . , Yn be independent, uniformly distributed random variables on the interval [0, θ].

a Find the density function of Y(k), the kth-order statistic, where k is an integer between 1 and n.

b Use the result from part (a) to find E(Y(k) ).

c Find V (Y(k) ).

d Use the result from part (c) to find E(Y(k) − Y(k−1) ), the mean difference between two successive order statistics. Interpret this result.

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) and Exercise 6.76 to find V (Y(k) Y( j ) ), thevariance of the difference between two order statistics.AnswerStep 1 of 1(a)We are asked to show that U = Y /Y and Y are independent. (1) (n) (n)We know the following theorem,\nLet Y 1 . . . , Yn be independent identically distributed continuous random variables withcommon distribution function F(y) and common density function f (y). If Y (k)denotes thekth order statistic, and j and k are two integers such that 1 j < k n, then the densityfunction of Y (j)and Y (k) is given byg (y , y ) = n! yj j1 yk yj k1j 1 yk nk 1 2, 0 y y (j)(k) j k ( (j1)! ×(k1j)!×(nk)!( ( ) ( ) ( ) j kLets apply the transformation U = Y (1)/Y(n) and V = Y (n), we have that y1= uv , y =nvand the Jacobian of transformation is v.Hence we can write, 1 n n2 f(u, v) = n(n 1) ( (v uv) v f(u, v) = n(n 1) 1 n(1 u) n2v n1, 0 u 1,0 v ( Since this joint density factors into separate functions of u and v and the support isrectangular,Hence Y (1)Y (n)and V = Y (n)are independent.