1. ## Conditional Expectation

Hi,
The problem is in the attached PDF file.

Thanks.

2. Originally Posted by ganiba
Hi,
The problem is in the attached PDF file.

Thanks.

RonL

3. Originally Posted by ganiba
Hi,
The problem is in the attached PDF file.

Thanks.
I'm not sure where this will go, but lets start by expanding the integrals, as $\displaystyle X$ and $\displaystyle Y$ are independent:

$\displaystyle E_Y\left[ E_X \left[ \chi_{\left\{ X>\alpha Y\right\}} X \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha y}^{\infty} x \ p_X(x)\ p_Y(y)\ dx dy$

and:

$\displaystyle E_X\left[ E_Y \left[ \chi_{\left\{ Y>\alpha^{-1} X\right\}} Y \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha^{-1} x}^{\infty} y \ p_X(x)\ p_Y(y)\ dy dx$

Now we can observe that the region over which we are integrating in the first integral is the part of the x-y plane below $\displaystyle y=\alpha^{-1} x$ and that in the second integral is that part of the plane above the same line.

Quite where to go from here I don't know

RonL

4. Originally Posted by CaptainBlack
I'm not sure where this will go, but lets start by expanding the integrals, as $\displaystyle X$ and $\displaystyle Y$ are independent:

$\displaystyle E_Y\left[ E_X \left[ \chi_{\left\{ X>\alpha Y\right\}} X \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha y}^{\infty} x \ p_X(x)\ p_Y(y)\ dx dy$

and:

$\displaystyle E_X\left[ E_Y \left[ \chi_{\left\{ Y>\alpha^{-1} X\right\}} Y \right] \right] =\int_{-\infty}^{\infty}\, \int_{\alpha^{-1} x}^{\infty} y \ p_X(x)\ p_Y(y)\ dy dx$

Now we can observe that the region over which we are integrating in the first integral is the part of the x-y plane below $\displaystyle y=\alpha^{-1} x$ and that in the second integral is that part of the plane above the same line.

Quite where to go from here I don't know

RonL
Thanks Ront.
I attached the file because i don't know how to include mathematical formulaes in the Forum, so thanks again for the help.

I want to add that X and Y are POSITIVE random distributions, so integral starts from 0 to infinity.

Thanks.

5. Originally Posted by ganiba
Thanks Ront.
I attached the file because i don't know how to include mathematical formulaes in the Forum, so thanks again for the help.

I want to add that X and Y are POSITIVE random distributions, so integral starts from 0 to infinity.

Thanks.
But that does not realy help, its a idea that's needed now.

ronL