Question

7. Show that if the joint probability density function of X and Y is if 0 < x <.. =sin(x + y) f(x, y) = { VI fres 9 Line + »» Hosszž, osys elsewhere, then there exists no linear relation between X and Y.

Answer 1

22 42 7) Given joint pdf of x and y as ferry) = | £ sintury) if os as the OSY STY elsewhere Covariance 13 the measure of linear relationship between two variable it is defined as cov (X,Y)= E(xy] - Elx) E[y] where E (XY]= lay ferrys dady E[x]- odx ** Etos - fjay for ydudiy E(X)= 1 rfeeds ECYD= Syfogsely for this we have to find marginal pdfs forandfey) we have fuse [fonisidy = , fex, yody у= 0

up comedy - 292 as Yue of (asont hus = my) [him - (h)uist = [rhxo) sm (h+%) sos) - 2 balconsofa xp Chteluisi 2.1. spinings I = Chat Groquers oto T2 i 7.50 + XUS - (um-2015-) - - {(42) -* + ) som V Rp chunussi

X (sinx tosk) de - Cson-tate de * ( suma de a avens die] lur= ufu-July +2 sina" + cose). ? = $(0-0 +(4-0) Thanay+0-1] (6] Similarly ECYD = lyteyrdy -" (sing teoryddy ** [ "anong day 4 Jahreny dls] T7 = $(9) Similarly from above we have just replace y by x) TE

Exy- I siniity) by dedy V2 TV2 | (sini wsy tasusing Jay dady V2Y Sin(A+)-sinf WB+LOSH Sin B ML TV2 y ddy | || cosesing rydudly we have from above AN zsinadu - y = 1 y voete opreme y come de Jasoyds = * fee £ 1 sepah dan u pa umea? *** [ {air (de + fjeme code] ML TV (4-0 TY2 (I-1) dx + Ixcole Codk YED ** [ 65+ )u) + (9-1)(0) = *(1-a] = -1

Hence COULX ,y) = ECxy] – E(X) Ely) Een ht - (HD = -0-04605 which is not nearear to ti or t. (covex,y) moj Hence there (exists) no relationship between xq y. (linearly). if couley is more then there existe relation