X~N(0,1), Y=X^2~[tex]\chi^2[/tex](1), find E(XY).(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Expectation of XY

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**