# Thread: Conditional expectation with inequality

1. ## Conditional expectation with inequality

Hi, so the definition of $\displaystyle E(g(Y_1)|Y_2=y_2) = \int_{-\infty}^{\infty} g(y_1) f(y_1|y_2) dy_1$ however how do we define $\displaystyle E(g(Y_1)|Y_2 \le y_2)$ ie, when the condition is an inequality rather than an equality?

Thanks

2. ## 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.

3. ## 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:

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

thanks

4. ## Re: Conditional expectation with inequality

Yes that looks right.