Question

10. Let X and Y have a discrete joint distribution with if (x,y) = (-1,1) P(X = 2, Y = y) = { = ; if x=y=0 = 0, elsewhere Find (a) the conditional distribution of Y given X = -1. (b) show that X and Y are uncorrelated but not independent. (C) Find the marginal distributions of X and Y.

Answer 1

Let & Y Soleelicon ico and have a discrete joins distribution with as S3/16 i (344) =(,1) P(X=*, Y = 9) = 144 x=y=0 Ottiesurse 0

According to le queesliay we qex Indºproba distribulion Discsele gourdom Tvariables X and Y Yl-1 a 2 O 3/16 0 3116 4/16 O 3/16 0 3116

(@ The conditional dispribution of y given x= -1 ЧЕ - ( P(Y= y(x=-1) = 0 с 9 = 0 — ГО 92 |

E (*)= {xp(x=) = -3 +0+ 3 - E(Y) = 3 P(y=4) = - 2 / 3 to + 2 この E&Y= Xy P(x=xy=4) = to me to 3/16 to +16 to toto I O

Then, cov (Y) = Exy)- EXECY) = 0-0 a o .: Con (XY) = 0 0 Bow equortion 0, we get X and Y ase incorrelated We have ? 3 P(x = 1, Y = 1) = 1 © p(x-1)=2 p(Y=1) = 3 Hesen p[(x=1) PCY=1) = (3) () = q 64

Бош кошиків 0 аад олода и се, P(x=1, Y= 1) = F(x-1) PC Y= 1) from equalion Wr we get xoud y ase not indepenout IV

The im X as ? imarginal distribution of 310 Ć x=-1 P(x=x) = x=0 나. 318 x=1 The mar marginal el distribution of Y as, 30 y=- | P(y=4) = 14 y=0 y = 1 318