Question

3.9. Problem*. (Section 9.1) The following problems concern maximums and minimums of collections of independent random variables. (a) Let Y.Y2, ..., Yn be independent exponential random variables with parameters 11, 12,..., In, respectively. Prove that E[min{Yı, Y2, ..., Yn}] < min{E[Y], E[Y2),..., E|Y.]} (b) Suppose that X1, X2, ..., X, are independent continuous random variables with uni- form distributions on (0,1). Compute E[min{X1, X2, ..., Xn}] and E[max{X1, X2,..., X.}]

Answer 1

Son as Y, 12,-- Yn indep. exp with parameter di, dz,-- th respectively. 9. Here Yin exp(di) . since min (Y,, Y2 --Yn) ~ exp (tit de tout dal 5 min (Yir Ys, --Yn) w exp, di) E ( min (9.8.-'m))štio EVYJDZI, E142] => ---, ElYnna tn © min ( E(Y), ECY) --- Elyn)). min I li could be any? to the so--In) - di 1 one from (1,2,3-ns] from 0 and ③ we can say that re - El min (Y, Y, --Yn)) < min( E(), Elya).-- Ecr») b) X,,X2 . - Xn ~uco, i provel Flx) = f 1. dt = x rth order statistes - fxr ) = (2-1)! (1.)! n-1 fxeu (u) = n. (1-2) yon n- fxo, (x) = h.x"

El Xaw) Su favo, ay din I a. n (1-1) dla n11 ci 2* n Ub 2:n f'x 24 Conight de Elxow) zn. Ble, nang E(X): 4*: x 2 da - - * - bio is not (***)] = npr El Xen) = Elmin (X,, --Xn)) = nB(2,6) Elx cm) = E(max (Xı --Xn]) = m | E( min (X, - - X»))= nB(2,1-1] El mar (*. -- X»))