Question

3. (50 pts) Let (X, Y) have joint pdf given by c, |y< x, 0 < x < 1, f(r,y)= 0, o.w., (a) Find the constant c (b) Find fx(x) and fy(y) and oyx (c) For 0 x 1, find fy\x= (y) and (d) Find Cov(X, Y) (e) Are X and Y independent? Explain why HYX=x

(a) Find the constant c.
(b) Find fX (x) and fY (y) (c)For0<x<1,findfY|X=x(y)andμY|X=x andσY2|X=x. (d) Find Cov(X, Y ).
(e) Are X and Y independent? Explain why.

Answer 1

a) lykt For valid PDF, 1.17 II fla, y) = 1 techedly = * ct 2x = t[a o act txtatafhdyt 39-(-x) = 21 for osast I For lyla ocas 2 4sat tagal for ky co for ozych Hyry)= fide = [2] = ty for teyro Hyyati chef* Inty Horosy</ Thus, fyly) = 1-lyl for oslykel | (c) fylleply)= f(x - 1 for oleh fx (2) a lykt and acnel = -144 <1 CamScanner

Date: Myxx - y. fylxx dy I dy 27 Larse | 6 xx = tly 1x=x] - My XXX - - - 3 - 0 ² - 2 a) Etxv)= |f ny dan daya La Liga Lo T E i ylltydy + sytty dy z ļ (y + y ) dy + Į (y-7) day = [] + [21:441)+(1-3) {[x] = { n. 24 cm = {2. 21 = 2 Cor [X,77= E [Xy] -ECxT ELYJE 0-1.2.3 Te) fx (x). fyly) = 2x. (ity) for Icyco - = 2x (lay) for ocycl Since f(x). f (y) & fxy (aby) X and you not independent. CamScanner