Question

Comparing two densities. Joint density (a) for random variables X and Y is given by: fxy(x, y) = 6e-23-if 0 <y<I<0. Joint density (b) for random variables X and Y is given by: fxY(I, y) = 2e -2- if 0 <1,7 <00. Fill in the following chart and determine whether or not X and Y are independent for both densities (a) and (b). fx() fy(y) EX EY EXY Cou(X,Y) Independent?

Answer 1

@ Salution: het XBy har fxy (2,4)=662-9 joint par lu<JCH 200 SI لے اڑی لال له fxra) = 5 x eyd = rens- ey x = 602 [-e "+] fxml= 6 (ex -3x) escala fald) = 6exed dn= 6ey [- en =30d [no te zy ] [ fyly) = 3 0 3 1 0 <q cos E(x) = Se Cha) de 186uern 3d den =6[ 7=65-$) :gentes E(X)= Erg) = syftgdy = 834 Say = 3 12 for TE(I) = - ] Erxy ) = ssay flagstay = honger & dady -65te da se zda-V12 Sezata tre] dy Eyed [- det gerne dj = 38 yey [-x - 22 de =38 yey [no-ot yet + t e zy} dy =38 223 24 dy = 3 [ 13 3 +2 12 = 3 [ + ] - + E(4) = 77 cor(x4 ) = E(X4)-ECX) EX4 ) = 7-&($) cor(x, y) = -2

Extras fary) =666m ) 3034 fard) f (y) & fxy priy) jenay are not independent 6 let fxy (ny) = 2 82X-4 Jungen fx) = 826 edh =202 [-ey you = 262 [-0 +2] I fon)= 26 27 no fy (17 = 8200 edde = 289 [-en To = [0th ] If (Y) = 6 ) y so fx (n). f (y) = 2021-y. Tfxral file) = fxy (219) ie x & 4 an independent. E(x) = xfxindan - Freda= 2 1 2 st TE(X) = Ert)= Syksy 19=5 yeddy = 12 h ECLI=1 E(XY ) = ErxECY) Sa: independents Elxy] = ON(XY) = E(XL) - EXZELL). = -Ź (1) Callxy)=0