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#### Dhamnekar Winod

##### Active member

- Nov 17, 2018

- 100

Let

[Tex] Z=1+X+XY^2[/Tex]

[Tex]W=1+X[/Tex]

I want to find

[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]

**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]

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