Assume we have two continuous variables $X,Y$; and we know the conditional probability density function: $f_{x|y}(x,y)$
How might we calculate the conditional probability over a range of values: $P(x_0 \leq x \leq x_1|y_0 \leq y \leq y_1)$
I know we could use the definition: $P(x_0 \leq x \leq x_1|y_0 \leq y \leq y_1) = \frac{ \int_{x_0}^{x_1}\int_{y_0}^{y_1}f(x,y)dydx }{ \int_{y_0}^{y_1} f(y)}$ where $f(y)$ and $f(x,y)$ are the probability distribution functions of the just $Y$ and then $X,Y$, the joint.
I can calculate $f(y)$ quickly, but I would rather not calculate $f(x,y)$. This is for a numerical application and calculating $f(x,y)$ is expensive. Is there a way to use the the conditional probability density function directly? I feel like the answer is a simple integral; but this is my first attempt at dealing with continuous and discrete probability density functions in the context of conditional/joint probabilities.
If someone has a good reference that helped one "graduate" from the low-level undergraduate style probability (nearly everything is discrete events) to working with continuous or discrete density functions directly, I would be most appreciative.
Best Answer
If you know $f_{x|y}(x,y)$ and $f(y)$ then
Define $h(x) = \frac{\int_{y_0}^{y_1}{f_{x|y}(x,y)f(y)}dy}{\int_{y_0}^{y_1}{f(y)}dy}$
$P[x_0 \leq X \leq x_1 | y_0 \leq Y \leq y_1] = \int_{x_0}^{x_1}{h(x)dx}$