Hi,

The problem is in the attached PDF file.

Thanks.

Printable View

- Jan 25th 2007, 08:14 PMganibaConditional Expectation
Hi,

The problem is in the attached PDF file.

Thanks. - Jan 25th 2007, 11:27 PMCaptainBlack
- Jan 26th 2007, 05:36 AMCaptainBlack
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:confused:

RonL - Jan 26th 2007, 06:50 AMganiba
- Jan 26th 2007, 06:59 AMCaptainBlack