Question

Problem 6: 10 points Assume that observable is a random variable W = min X, <i<5 where {X; : 1<i<5} are independent and uniformly distributed on the unit interval (0 < x < 1). 1. Derive CDF for W, that is F(w) = PW <w]. 2. Evaluate density function, f(w) and identify it. 3. Find expected value of W. 4. Determine variance of W.

Answer 1

TOPIC:Uniform distribution and order statistics.

7 Problem - 67 here, {x; : Isis 5} are ind uło, random variables. ie the common pdf of xi's are- tx: (a) a § ! ;osxsl. 0 ; otherwise. in The common cdf of xi's are- 27 Fx; (X) = P(x: 5x). S . (9 d. = x ; osxsl. Ther, Fx; (x) = { o ;XCO. oExcl. ; xyl,

Define, w = min xi = x1, say, 1 <i5 i caf of w is y F (W) =P (www). = P(x n sw). F: Xayw.si 1-P (x0, xw, 2) All the Xi's are yw = 1-1 (x, yw, x 2 YW, _ , Xg? W) & 1216115 I due to independence [ace the protagon. * *67). [ t,(a= + (43) (Oy - i) - (-w). a 1-(1-w) s joswan Thus, the caf of w is - F(W) = o ; w co 11- (1-w) s josusli II ; wyli

2x so, the pod of wisa =) dw) = . [ F (w] * + 5 (1-w) 54. 1w) (= 5 (1-w)" ; 0 5Wsi. I = 0 ; otherwise. - O which is Beta C the distribution- [ Reason - suppose XN Beta (m, n). Then, the pdf of x is no deres. sant B (m, n) - ; 05231 ; ele. ; then, jx) Put, m=1, n = 5 reduces to - -) MX1 BUST (1-x)';05 KS I. B (m, n r (m) 1(n) T: B(m, n) - nemocens : B(1,5) = M r. (m+n) 5 (1-40)*:05x5! lao elsewhere. ioanal. Loco (

37 We needs => E (W) w how) due. = 5S wou)" dus. VO [B(m, n) = 5. B (2,5). = xm-1 (1-x)"4dx r(2). 8 (5) (17) L and, B(m, n) 5x 18 x 4! - s(m). Mm) 61 r(men) 5. - 4. Also, we have E (W²). = ('w 2 Hw) dw, 2 550 w2 (1-w)" dw.

5. (..)". - 5. B (3,5). - 5. (3). r (5) 118) (3-D: * (5-)! (8-)! 12 (1) = (-1); Jon, kt It = 5 x 2!*" ! 71 Thus, var (w) = E (W2) – E²(W). - - (+)? 252 = 0.019841. Tansh