Question

Let X1, X2, ... be independent continuous random variables with a common distribution function F and density f. For k > 1, let Nk = min{n>k: Xn = kth largest of X1, ... , Xn} (a) Show Pr(Nx = n) = min-1),n>k. (b) Argue that fxx, (a) = f(x)+(a)k-( ++2)(F(x)* (c) Prove the following identity: al= (+*+ 2) (1 – a)', a € (0,1), # 22. i

Answer 1

to see and Solution: a) Let Ai denote The event that Xi is the the largest of X1,..., X; It is is easy that these ere independent events P(A ;) = 1/2 P{ Na an} = PLA A A A K-1 kt k K-1 n(n-1) n-2 n- kt no 6) Since knowledge of the set of values & X. , . . . , Xn} gives us information about the order of these random variables it follows that given Nu=n, the conditional distribution of XNk is the same the distribution of the lith largest of random variables having distribution F. as

Hence, make the change of variable in-k ExN (X) = n=k nin-D) {n-k)! (k-1)! X{F()** (Fr.) 400) kol n! using induction, when u=2 a Što 460-63 1) ОА 3 (1) (-a)! assume it for k. be giren and suppose (1) is true for let we meR be m=k. then =Ž citat ) 4-223" 더 TE 04 By induction hypothesis of is proved = (tk) (1-2 - .k=m+1, 1-4 the hoolds proved a 120