Question

3. Let X and Y have a discrete joint distribution with Table 1: Joint discrete distribution of X and Y Values of Y -1 0 1 Values of X -1 1 į 0 1 1 0 -600-100 Then, find the following: • the marginal distribution of X; [2 points) • the marginal distribution of Y; [2 points] the conditional distribution of X given Y = -1; [2 points] Are X and Y are independent? Discuss with proper justification. (3 points) • Cov(X,Y). [2 points]

Answer 1

DATE: Solution in We have given, the joint discrete distribution of x and y. total - IT Y 218 o Y8 total [ 218 618 1. ET Marginal distribution of The marginal probablity distribution of X -88 11. Total prx=x) 218 6/8 * Marginal distribution of Y: I The Marginal probablity distribution of T- 1 0 Tobel Ipry=y) 5/8 28 78 1.

* The conditional distribution of a given Y=-1 is, P(X=-1 Y=0-1) = P(X=-1, Y=-1) Ply=-1) 18 518 P(x = 0 -1 | x=-1) = / P(x=1 / x=-1) = P(X= 1, Y=-1) PCY=-1) Y2 518 P(X=1 | Y=-1) = 4 Conditional diston probablity - distribution of Y=ot is p(x=x) uls

are independent ? * Are x and r justify The to show joint X and Y probablity distobution (x, y) are independent. 1 P (x;) X - Y8 Y2 Y2 518 o o X4 218 ТУg oys ply) 248 618 kle see that, a p(x=-1, Y = -1) = /g and P ( X = -1). P(X=-1) = 2 x 5 - 10 P (X=1 , Y =o) = 0 and PCX=-1.p6Yao) = 2 x 2

DATE p(x= -1 , Y = 1) - and p(x=-1). Pry=1) = 3 . b - 2 3 and similarty ke can see that the probablity in each cell is not equal to the product of the corresponding marginal probablities on the right and bottom, Hence, X, Y are not independent. (3 (1- (1-3 (1) coyex,y) + 0 0 We know that, cor (X,Y)= E(XY- ECX), ECY). EX) = - + e E(X) = 4

PAGE NU. DATE: ECY)= -L. + 0 +1. EcxY)= E Pij Xi Yj ExY) = 1/2 (-1) (-1) + 1 / 2 (-1) (1) - 40 tot 7 (1) (-1) Y x x EXY) = 401 - 15 EXY) = -4/8 % EXY) = - %

PEACOCK PAGE NO DATE: Here, covex)= Ecx y) - E(X) LYO. CON(XY) = -% - 4/84/ corex,y) = -Yn- te Covex X) = -32-15 64 Cor(x, y) = - 48 6LP CONCX,Y) =