Exercise 5.6.12. Suppose Yi, ,Yn are independent Exponential(1)\'s, and Y(1) Yn)
ID: 2949223 • Letter: E
Question
Exercise 5.6.12. Suppose Yi, ,Yn are independent Exponential(1)'s, and Y(1) Yn) are their order statistics. (a) Write down the space and pdf of the single order statistic Ya). (b) What is the pdf of Yuy? What is the name of the distribution? Exercise 5.6.13. The Gumbel() distribution has space (-0, 00) and distribution func- tion (5.103) (a) Is this a legitimate distribution function? Why or why not? () Suppose X..X are independent and identically distributed Gumbel(u) random variables, andY- Xn), their maximum. What is the distribution function of Y? (Use the method in Exercise 4.4.18.) (c) This Y is also distributed as a Gumbel. What is the value of the parameter? The remaining exercises are based on .., independent Uniform(0,1)'s, where Y = (u(1), . . . , u(") is the vector of their order statistics. Following (5.86), the order statistics are be represented by Uo Gi+G, where G (GG) Dirichlet(a1 , a2, . . . ,?? +1 ), and all the ai's equal 1. Exercise 5.6.14. This exercise finds the covariance matrix of Y. (a) Show that (5.104) 80 Chapter 5. Transformations: Jacobians where I, is the n x n identity matrix, and 1, is the n x 1 vector of 1's. [Hint: This covariance is a special case of that in Exercise 5.6.9(d).] (b) Show that Cov[Y] = (n + 1)(n + 2) (B-n+1 6109) (5.105) where B is the n × n matrix with ijth element bij min{i,j), and e-(1,2, , ny [Hint: Use Y-GA, for the A in (5.87), and show that B = AA, and c = A1".] (c) From (5.105), obtain i(n+1- (5100)Explanation / Answer
(5.6.12)
(a) Let Y1,Y2,... ,Yn be n random variables. Then the corresponding order statistics are obtained by arranging these n Y(k)'s in nondecreasing order, and are denoted by Y1:n,Y2:n,...., Yn:n. Here, Y1:n, is the first order statistic denoting the smallest of the Y(k)'s, Y2:n is the second order statistic denoting the second smallest of the Y(k)'s, ..., and Yn:n is the n th order statistic denoting the largest of the Y(k)'s.
It is important to mention here that though this notation for order statistic is used by most authors, some other notations are also employed in the literature.
(b) The pdf of Y(1) is = x ? D : P(X = x) ? 1 and it is a continuous distribution form right.
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