The random variable pair X Y has the joint distribution Find

The random variable pair (X, Y) has the joint distribution Find: The conditional probability P(X|Y = 2). The covariance Cov(X, Y). Are X and Y independent? Justify your answer.

Solution

7. the marginal distribution is obtained as

P[X=1]=1/12+1/6+0=1/4                          P[Y=2]=1/12+1/6+1/12=1/3

P[X=2]=1/6+0+1/3=1/2                            P[Y=3]=1/6+0+1/6=1/3

P[X=3]=1/12+1/6+0=1/4                          P[Y=4]=0+1/3+0=1/3

a) we need to find the conditional probability of P[X|Y=2]

now P[X=1|Y=2]=P[X=1,Y=2]/P[Y=2]=(1/12)/(1/3)=1/4

P[X=2|Y=2]=P[X=2,Y=2]/P[Y=2]=(1/6)/(1/3)=1/2

P[X=3|Y=2]=P[X=3,Y=2]/P[Y=2]=(1/12)/(1/3)=1/4 [answer]

b) cov(X,Y)=E[XY]-E[X]E[Y]

now E[XY]=1*2*1/12+1*3*1/6+1*4*0+2*2*1/6+2*3*0+2*4*1/3+3*2*1/12+3*3*1/6+3*4*0=6

E[X]=1*1/4+2*1/2+3*1/4=2       E[Y]=2*1/3+3*1/3+4*1/3=3

hence cov(X,Y)=6-2*3=0 [answer]

c) No, X and Y are not independent.

because if X and Y are to be independent then P[X=i,Y=j]=P[X=i]*P[Y=j] for all i and j

but here P[X=2,Y=3]=0   but P[X=2]*P[Y=3]=1/2*1/3=1/6 but not zero.

hence P[X=2,Y=3] is not equal to P[X=2]*P[Y=3]

hence they are not independent [answer]