I'm reading about statistical decision theory and started to wonder how would you write conditional probability in terms of density function?
For example, if we have random variables $X$ and $Y$ then we know that:
$$P(Y, X) = P(Y \;|\; X)P(X)$$
Now lets say $X$ and $Y$ have continuous range so if I would like to calculate the joint probability that $X$ is in range $[x_1, x_2]$ and $Y$ is in range $[y_1, y_2]$ I would do:
$$\int_{x_1}^{x_2}\int_{y_1}^{y_2}f(x,y)\:dy\:dx$$
where $f(x,y)$ is the joint probability density function of the two variables. Now my first question is: Is is this notation correct?:
$$P(Y\; \text{in range}\; [y_1, y_2], X\; \text{in range}\; [x_1, x_2]) = \int_{x_1}^{x_2}\int_{y_1}^{y_2}f(x,y)\:dy\:dx$$
If yes then my next question is: IF:
$$P(Y, X) = P(Y \;|\; X)P(X)$$ $$=>$$
$$P(Y\; \text{in range}\; [y_1, y_2], X\; \text{in range}\; [x_1, x_2]) $$$$= P(Y\; \text{in range}\; [y_1, y_2] \;|\; X\; \text{in range}\; [x_1, x_2])P(X\; \text{in range}\; [x_1, x_2])$$ $$= \int_{x_1}^{x_2}\int_{y_1}^{y_2}f(x,y)\:dy\:dx$$
Then how do you write $P(Y\; \text{in range}\; [y_1, y_2] \;|\; X\; \text{in range}\; [x_1, x_2])$ and $P(X\; \text{in range}\; [x_1, x_2])$ in terms of the density function? (So that the expressions include integral signs, dx, dy and f(x,y), etc.)?
Hope my questions is clear and understandable. If not please let me know :)
Thank you for any help :)
$$ P(Y \in [y_1, y_2] \mid X \in [x_1,x_2]) = \dfrac{P(X \in [x_1,x_2], Y \in [y_1,y_2)}{P(X \in [x_1,x_2])} = \dfrac{\int_{x_1}^{x_2} \int_{y_1}^{y_2} f(x,y)\; dy\; dx}{\int_{x_1}^{x_2} \int_{-\infty}^\infty f(x,y)\; dy\; dx}$$