Question

Problem D: In each part, find the covariance and the correlation of X and Y and interpret the correlation value (xy2 1), x 1,2,4, y = -1,2 a) the joint pmf of X and Y is p(x, y) 41 b) the joint pdf ofX and Y is f(x, y) = for 0 yx< 1.

Answer 1

á la mise en la red eyes w w w www (a) Ginen joint proof of & and I, - P(xx, lag), 1 (2y²+12 , x= 1, 2, 4 - y = -1,2 which can be tabulated as T X 2 (-3) >> Marginal density of y. T I TA % "74)3741/ P(x=x) 141 144 2, Marginal density of x From the table we can easidy Calculate Ex), ECXS VOX EXY2EC42) vey2 and Elly2: Coversi come cov (XY) - ÉcxY2 - Ecx) ECY) ECD- &* P(x2x) = 1*74+82x1.5 +++22, 2 ay Elx²)= S x² P(X) = 147/41+ 2x12% +44 22 = 407 1. Vacarice, V(x)= E(XL)–(EX)) = 107,91 ing. 16487-tung 3 2524/1081 – 1.5027 EX") - SS xy PCXap, tay) -(160) *#+ E172) +(1242 +230 +x23%+| _274 (17/ ** *4* - *- 34561 +7 41441. 412

Ev). . y Plyoga - 728 Competenzen Ecra)gArmy)= (Its - " * varavie, viN)= E(Y3) EVO2 = 134. sz., 542-2804_2740 VAN nue 5494-2704 2790 1661 1687 T.6597 Now covariance con(x,y) = E(XY)-F(x) Ely) = 1544 - 1952 71x4 e 6314-6188_ 126_- 1681 corelation (814)- CONCXY) JVCD) July 1661 - - Now 12/2 126 50.2593x 52.8204 26 - 0.04746 50.2593 x 52.7209 Since comelation coefficient is near to o we can say there is a little conelation between X and Y.

by joint pdf of X andy, feasy) = x, ocy exch Marginal densing & fexsypdy - It dy at (yan - fx (@zh Öcx</ Marginal density of fyly)= ffexsyp dx = 6 + 2 dx loge * 9 (log 1-logy) -logy 3 log ý log l=0 . .. : - dute log oxy si e log y , ayat Elx)= J x fexde Saxidx = f (x3 = ½ E6x3* Lifeso dx fide fils Yell Lino VORS= £172) EC4)]=4 - / -1/2 = 1/2 E97= fgy fagyly z fy colpay - Pylogg dy vitin Zo flogy ny dy]

- ਦੇ - ਨੂੰ , (461 147 ... | =0 : 4 Ely2)Ty? Fazyn dy =" Sy -tag (7)dy = -tytog ydy 9x9 95 ਰੰਧ, 0449 -ਤੇ 94 ) 4les - 4- 4 4 ਤੇਲr --07. . V(52. &੫੭-੫)))- -- 3 - M.

ElxY)= { [zy feay)dy da. ocy <m</ 1, 2 2 A 1402 - = “ įydy dx = $3.45%dx_ fxy / dy da ೨೭ Hydyd 290 yao oo cov (8,4)= EL XV)– Elt) ECY)= “ tatt Chat - 30* - Now comelation ayo= CNCxiy Vich J De 1/24 -2.44 F1704 0.287 Xoideos so417 0.0637 0.6576 - Here we have a more stronger correlation between X and Y them the previous one