# Conditional Expectation

• Jan 25th 2007, 09:14 PM
ganiba
Conditional Expectation
Hi,
The problem is in the attached PDF file.

Thanks.
• Jan 26th 2007, 12:27 AM
CaptainBlack
Quote:

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

Thanks.

RonL
• Jan 26th 2007, 06:36 AM
CaptainBlack
Quote:

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 $X$ and $Y$ are independent:

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

and:

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

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 $y=\alpha^{-1} x$ 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:confused:

RonL
• Jan 26th 2007, 07:50 AM
ganiba
Quote:

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

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

and:

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

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 $y=\alpha^{-1} x$ 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:confused:

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.
• Jan 26th 2007, 07:59 AM
CaptainBlack
Quote:

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