below is joint probability distribution of X and Y:
X | ||||
Y | 0 | 1 | 2 | Total |
0 | 0.2000 | 0.2000 | 0.2000 | 0.6000 |
1 | 0.1000 | 0.1000 | 0.1000 | 0.3000 |
2 | 0.0000 | 0.1000 | 0.0000 | 0.1000 |
Total | 0.3000 | 0.4000 | 0.3000 | 1.0000 |
a)
marginal distribution of X:
X | P(X) | XP(X) | X^2P(X) |
0 | 0.3000 | 0.0000 | 0.0000 |
1 | 0.4000 | 0.4000 | 0.4000 |
2 | 0.3000 | 0.6000 | 1.2000 |
total | 1.0000 | 1.0000 | 1.6000 |
E(X) | = | 1.0000 | |
E(X^2) | = | 1.6000 | |
Var(X) | E(X^2)-(E(X))^2 | 0.6000 |
marginal distribution of Y:
Y | P(Y) | YP(Y) | Y^2P(Y) |
0 | 0.6000 | 0.0000 | 0.0000 |
1 | 0.3000 | 0.3000 | 0.3000 |
2 | 0.1000 | 0.2000 | 0.4000 |
total | 1 | 0.5 | 0.7 |
E(Y) | = | 0.5000 | |
E(Y^2) | = | 0.7000 | |
Var(Y) | E(Y^2)-(E(Y))^2 | 0.4500 |
b)
here as P(X=0)*P(Y=0)=0.3*0.6 =0.18 ; which is not equal to P(X=0,Y=0)=0.20 ; hence X and Y are not independent
c)
expectation of X =E(X)=1
expectation of Y =E(Y)=0.5
Variance of X =Var(X)=0.6
Variance of Y =Var(Y)=0.45
d)
here E(XY)
=xyP(x,y)=0*0*0.2+1*0*0.2+2*0*0.2+0*1*0.1+1*1*0.1+2*1*0.1+0*2*0+1*2*0.1+2*2*0=0.5
hence Covariance =Cov(X,Y)=E(XY)-E(X)E(Y)=0.0000
therefore Correlation =Cov(X,Y)/(Var(X)*Var(Y))1/2 =0
as correlation between X and Y is 0 ; threfore X and Y are uncorrelated
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