Question

8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .

Answer 1

Answer: Criven That :- X,,ta.... Xn wo (0.1) random variables let let. Mno mar (x1,x2,.--*n) Now) Now we know that Pdf of geth ader statistics fuers) = Y- ni (8-1)] (n-v); (Flu))' (1-660** PQ f(x) → Cof of f(x) rn, XcnJ= M. Flu); n(f(x))"77() now, integrate on it f(x)= k lidt = t ax

fava nx4 Now. in is substitute XenyM then *(M) = nmnt for olmal Now, - Now a show that Probability to is Mnal I as n Min. Converges in that o Nowo we have E (Mn) S'm. nondo com s in forma dom Ens ma com

Now, FCmn de tot E (Mn) = 1 then, Mr B E (Mn) as not T Mr L l as non as nosa → Show that nCl-mm) Expli) Converges in distribution to an with mean 1 as nyn that is n(1-Mn) Exponential av .. we have - nm n for ocmai for oz o, otherwise 4=n(1- An)

Now, F[Y): P[Y29) = P. [n(1-m) sy) = PC (-m 34 mi) = PC m 21-) I n m mnam - na comm), un dm : n on fly) - 1 (in take Fly): esting Fly]: 0-(1- to jo= (-4), Now, apply the limits then, we will get lim fly )= lim linq gn- i nete hu y is exp(1) if a Nexp (1) then it will happen for 9-n (1-rin) follow Expli) where noso