Question

4. (30 pts) Let (X,Y) have joint pdf given by < , | e-9, 0 < x < f(x,y) = 3 | 0, 0.w., (a) Find the correlation coefficient px,y: (20 pts) (b) Are X and Y independent? Explain why. (10 pts)

Answer 1

) = 8* ,9) dxdly y b v v ryba - jx}jve" by idx ہے اس =3 2 + 5 عه به ريه علي ول ل - وهه له } * ل و لا لا لا - ۶۶ نے فعل = لاتے (هالي = = ه = 4 پوندو ته = (۶) ها بود و ز = (۲) دی : رستے سے و-4 1 = 3 = *ع : ج 1 * 9 1= . د : 3 - :eq-3 =2 در دیده أي فعل = فيه عه (وه لاته و مع ها (هه ل وه ل - ودول عد} = عه بوده و در 2 + و - ي= .ه) مه ولد *eع في : ا 25

Now I sey dy = {y verydy - Joey ’dny dyre s +ly é* - E(+²=9 Relax to getén ax la +2 Ixé*data ſetax 2 + 2x 12 + 2x1-6. Ć COVcx,y) = E(XY) - ELX, ECT) = 3–1*2=1 var (x) = E(x²,-E²(x)=2-1=1 var (T) = ELY, -E²(Y) = 6-23=2. s p(x,y) = ya. (6) we find that the range of (2,9%) depends - on each other in the joint PDF. I. xty can not be independent. Again Per Pix,7) = to implies that we y are dependent.
the correlation is obtained without deriving the marginals, only definition of expectation is used. For further query in above, commrnt.