- #1

gimmytang

- 20

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X~N(0,1), Y=X^2~[tex]\chi^2[/tex](1), find E(XY).

My thoughts are in the following:

To calculate E(XY), I need to know f(x,y), since [tex]E(XY)=\int{xyf(x,y)dxdy}[/tex]. To calculate f(x,y), I need to know F(x,y), since f(x,y)=d(F(x,y)/dxdy.

[tex]F(x,y)=P(X\leq x, Y\leq y) \\

=P(X\leq x, X^{2} \leq y)\\

=P(X\leq x, -\sqrt{y} \leq X \leq \sqrt{y})[/tex]

Thus,

[tex]F(x,y) =P(-\sqrt{y} \leq X \leq x)P(x<\sqrt{y})+P(-\sqrt{y} \leq X \leq \sqrt{y})P(x > \sqrt{y})[/tex]

Then I don't know how to calculate the four components of probabilities accordingly. Anyone gives a hand?

Thanks!

gim

My thoughts are in the following:

To calculate E(XY), I need to know f(x,y), since [tex]E(XY)=\int{xyf(x,y)dxdy}[/tex]. To calculate f(x,y), I need to know F(x,y), since f(x,y)=d(F(x,y)/dxdy.

[tex]F(x,y)=P(X\leq x, Y\leq y) \\

=P(X\leq x, X^{2} \leq y)\\

=P(X\leq x, -\sqrt{y} \leq X \leq \sqrt{y})[/tex]

Thus,

[tex]F(x,y) =P(-\sqrt{y} \leq X \leq x)P(x<\sqrt{y})+P(-\sqrt{y} \leq X \leq \sqrt{y})P(x > \sqrt{y})[/tex]

Then I don't know how to calculate the four components of probabilities accordingly. Anyone gives a hand?

Thanks!

gim

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