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
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
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.
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