- #1
WMDhamnekar
MHB
- 378
- 28
Let X and Y be two independent [Tex]\mathcal{N}(0,1)[/Tex] random variables and
[Tex] Z=1+X+XY^2[/Tex]
[Tex]W=1+X[/Tex]
I want to find Cov(Z,W).
Solution:-
[Tex]Cov(Z,W)=Cov(1+X+XY^2,1+X)[/Tex]
[Tex]Cov(Z,W)=Cov(X+XY^2,X)[/Tex]
[Tex]Cov(Z,W)=Cov(X,X)+Cov(XY^2,X)[/Tex]
[Tex]Cov(Z,W)=Var(X)+E(X^2Y^2)-E(XY^2)E(X)[/Tex]
[Tex]Cov(Z,W)=1+E(X^2)E(Y^2)-E(X)^2E(Y^2)[/Tex]
[Tex]Cov(Z,W)=1+1-0=2[/Tex]
Now E(X)=0, So [Tex]E(X)^2E(Y^2)=0[/Tex], But i don't follow how [Tex]E(X^2)E(Y^2)=1?[/Tex] Would any member explain that? My another question is what is [Tex]Var(X^2)?[/Tex]
[Tex] Z=1+X+XY^2[/Tex]
[Tex]W=1+X[/Tex]
I want to find Cov(Z,W).
Solution:-
[Tex]Cov(Z,W)=Cov(1+X+XY^2,1+X)[/Tex]
[Tex]Cov(Z,W)=Cov(X+XY^2,X)[/Tex]
[Tex]Cov(Z,W)=Cov(X,X)+Cov(XY^2,X)[/Tex]
[Tex]Cov(Z,W)=Var(X)+E(X^2Y^2)-E(XY^2)E(X)[/Tex]
[Tex]Cov(Z,W)=1+E(X^2)E(Y^2)-E(X)^2E(Y^2)[/Tex]
[Tex]Cov(Z,W)=1+1-0=2[/Tex]
Now E(X)=0, So [Tex]E(X)^2E(Y^2)=0[/Tex], But i don't follow how [Tex]E(X^2)E(Y^2)=1?[/Tex] Would any member explain that? My another question is what is [Tex]Var(X^2)?[/Tex]
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