Question

7. A service facility operates with two service lines. On a randomly selected day, let X be the proportion of time that the first line is in use whereas Y is the pro- portion of time that the second line is in use. Suppose that the joint probability density function for (X,Y) is f(tv) = ( +y), if 0 <r,y<1. elsewhere. (a) Compute the probability that neither line is busy more than half the time. (b) Determine whether or not X and Y are independent. (c) Find E(X+Y) and E(XY). (d) Find o, o, and the covariance OXY. (e) Find the variance of X+Y, that is, o+y: (f) Find the correclation coefficient Pxy.

Answer 1

Sch". ② The joint pelf for (x,y) is flony) = 3 (x² + y2), if osnyal 1 o else where Probability that neither line is busy more than half the time = P(x< < 1 YCZ) = shsh 3 (x² + y²) dundy - $ fe y + y dre stajale = S ]. - ?<H+te) = { x = 6 N

③ The marginal p.dif. of x is falu = s'fmosdy = f(x?tygody =} (x+) oxuc and marginal p.d.f. of y is fo(t) = ('ffy) dx = 3 x + y dn = 2(x + y ² ) = ? (y² + 7) , olyai Since fron) fulds & fxylbyl so X and Y are not independent. © E(X+Y) = E(X) + f(1) = s'afalda & fy fylgsdy = 36 (2 + 4) dx + 2 S (y + dy E(XY) = S's 'ny f6ys dredy ==S D' xy. (o'tg) andy = 38. [5.4633 + xg) a7.de = 1 + i) dm = 1x x = Varlx) = E(X²) -(Ex)] = f. xx (x + 1) de J! - Amy ( 3 ) = 1/3 - 235 - - 83 / Vax(x) = varlys Also y = E) - (x - 1 y ² x 2 (y2 + # ) dy - 1 Cow X,X) = Oxy - E(X4) -EX)EGY) = {x{ = - @ Var(x+4)= Varlx) + Var(4) + Coux e - Cor(x,y) Vary) Svartti And Cou(x,y) Varlx) -1.86 Any = 0.0169 Ang