# Conditional expectation with inequality

• February 13th 2013, 07:39 PM
usagi_killer
Conditional expectation with inequality
Hi, so the definition of $E(g(Y_1)|Y_2=y_2) = \int_{-\infty}^{\infty} g(y_1) f(y_1|y_2) dy_1$ however how do we define $E(g(Y_1)|Y_2 \le y_2)$ ie, when the condition is an inequality rather than an equality?

Thanks
• February 14th 2013, 03:19 PM
chiro
Re: Conditional expectation with inequality
Hey usagi_killer.

For the inequality condition what is done is we find a random variable that fits the definition of Y2 <= y2.

The easiest way to do this is have a double integral with the limits of Y2 going from -infinity to y2.

When you evaluate this double integral you will get an expression in terms of y2.
• February 14th 2013, 06:18 PM
usagi_killer
Re: Conditional expectation with inequality
Thanks chiro, so what would the expression look like? Do we still use the conditional density function?

Would it be:

$E(g(Y_1)|Y_2 \le y_2) = \int_{-\infty}^{y_2} \int_{-\infty}^{\infty} g(y_1) f(y_1|y_2) dy_1 dy_2$

thanks
• February 14th 2013, 08:12 PM
chiro
Re: Conditional expectation with inequality
Yes that looks right.