Question

Problem 10: 10 points Assume that a sample {X;:15; <4} of size 4 is drawn from the uniform distribution Unif(-1,1). Consider the maximal order statistic, X(4). 1. Derive density function of X(4) 2. Evaluate expectation of X(4) 3. Determine variance of X(4)

Answer 1

TOPIC:Distribution of the maximal order statistic.

Problem -10% Here, {x; : 14; 44} be a random sample of size 4, from w(-1,D distribution So, the pdf of x; is - x; (x) = { t = 1; -12 X 511 -122sl. 1-1 otherwise. . The adf of x; is - =Y F7; (x) = P(x; £x). da; (4) dt. = x+; -15 851. ire .(W) = o ; x<-1 Txt ; 1. 1 -14X ; xxi. *j = 1,2,3,4,

D Define, tu = max. { x 1, X2, -, Xu). So, the edf of tens is - * Fx (x) = P(xcw, 32). tew ex = P ( x, EX , X₂ EX , - , X 4 22 - 4 p (x; & x). Ки) =) Al the x;'s are sx; * j = 1004 [Fx; (x) = x+1 = 17 Fx; (x) . 2 jag Jos, -IERSI - (ME) ; -14X21. Go, Fx (x) = o ; X<-1 2 / 4 -1=2s1. ixxl. So, the body of this -> dx (4) ELEK, ( [). (2 . (2x)"

So, So, Ido (x) = (x+13 10.x4 -12 X El. otherweise (Ans 2X We need = E(X), - l'ad, (x) dx . *S (2=3° de * *(2843*** 32-) de (****** **) de ****** * = [(+ 4 +1+1) -66+-+7

3ix we need, var [Xew L' den body - + [' *(4) de +(\** (62+ 32 + 32-) de -|(2++32 + 3+ + 2) +(94*** -1 -

So, var [X4) [x2] – ² {xon). 0.106 75 Tans