Conditional Expectation

**ganiba** · January 25th 2007, 08:14 PM

Hi,
The problem is in the attached PDF file.

Thanks.

**CaptainBlack** · January 25th 2007, 11:27 PM

Originally Posted by ganiba

Hi,
The problem is in the attached PDF file.

Thanks.

I hate having to follow a link to see the problem.

RonL

**CaptainBlack** · January 26th 2007, 05:36 AM

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

RonL

**ganiba** · January 26th 2007, 06:50 AM

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

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.

**CaptainBlack** · January 26th 2007, 06:59 AM

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

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