Question

# Consider the following joint distribution between random variables X and Y: Y=0 Y=1 Y=2 X=0 P(X=0,...

Consider the following joint distribution between random variables X and Y:

 Y=0 Y=1 Y=2 X=0 P(X=0, Y=0) = 5/20 P(X=0, Y=1) =3/20 P(X=0, Y=2) = 1/20 X=1 P(X=1, Y=0) = 3/20 P(X=1, Y=1) = 4/20 P(X=1, Y=2) = 4/20

Further, E[X] = 0.55, E[Y] = 0.85, Var[X] = 0.2475 and Var[Y] = 0.6275.
a. (6 points) Find the covariance between X and Y.
b. (6 points) Find E[X | Y = 0].
c. (6 points) Are X and Y independent? Mathematically justify your answer.

from given data:

a)E(XY) =xyP(x,y) =0*0*(5/20)+0*1*(3/20)+0*2*(1/20)+1*0*(3/20)+1*1*(4/20)+1*2*(4/20)=0.6

therefore Cov(X,Y) =E(XY)-E(X)*E(Y) =0.6-0.55*0.85 =0.1325

b)P(Y=0) =5/20+3/20 =8/20

E(X|Y=0) =x*P(X|y=0) = (1/P(Y=0))*x*P(x,y) =(1/(8/20))(0*(5/20)+1*(3/20))=3/8 =0.375

c)

since P(X=0) =5/20+3/20+1/20 =8/20

P(Y=0)=8/20

P(X=)*P(Y=0)=(8/20)*(8/20)=0.16 which is not equal to P(X=0,Y=0) =5/20 , therefore X and Y are not independent

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