Question

Problem 5 Let Y1 denote the minimum of a random sample of size n from a distribution that has pdf f(x) e(,0x< o0, zero elsewhere X- n (Y1 0), find the cumulative distribution function (cdf) for Zn = n (Y1 - 0), and Let Zn find the limiting cdf of Zn as n >oo.

Answer 1

Given that:

f(x) = e^-(x-$\theta$),$\theta$<x<∞

Let Zn = n(Y1 - $\theta$ )
Answeilir let y, denote the minimum of a random sample of size n from a distribution with probability density bunction ocUCD elsewhere. The distribution bunction is Fala) = P(XS) Fa (4) - 1949) fy(V)=fé (4-0) du fo(a)=-4 (0-0) (-20) fulua -(8-0), x(u) ali (220)

The cumulative density function of 4, is Fy Lt) = priminę x,,X2 -.. Xny st) Fyrlt) = 1 - rv (Xizt, Xzzt....Xn2t) fylte 1-P prevoza) X Proxy?t)*---* py(xmat) Fylt) =1-Core Xe 2t)) Fylt) 21-11-prexy st)) Fy, (t) = 1=(1-FX+ (t))" fylt)=1-11-(18-16-oggen fuilt) = tft te togn Fuilt) = nie net-0) ell Zn =(4,-0) the distribution of Ln wil Lalth = p (Int) Enll) - pln(4,-0) st) Zn(e) = P(4,5 tonto) Inlt) = f4, [Ento]

Zn(t) zle nento)-o) Zn(t)= ten ( 4 16-0) Zalt) = 15et, txo the limiting distribution of mones F2 (6) = llim Ey it all-et, to 2n To else where nd