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) =0.60.55*0.85 =0.1325
b)P(Y=0) =5/20+3/20 =8/20
E(XY=0) =x*P(Xy=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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