Is this the definition of conditional probability of 2 continuous random variables conditioned on 1 continuous random variable? Please provide a reference.
(I forgot if any 2 continuous random variables necessarily have a well-defined joint pdf.)
Instead of the usual $$P(z_1 < Z < z_2 | B=b) := \int_{z_1}^{z_2} f_{Z|B=b}(z) dz,$$ with $f_{Z|B=b}(z):=\frac{f_{Z,B}(z,b)}{f_B(b)},$ it looks like we'll have like $$P(z_1 < Z < z_2, u_1 < U < u_2 | B=b) := \int_{u_1}^{u_2} \int_{z_1}^{z_2} f_{(Z,U)|B=b}(z,u) dz du,$$ with $f_{(Z,U)|B=b}(z,u) := $, I think, $\frac{f_{Z,U,B}(z,b,u)}{f_B(b)}$
- Note: if any of the definitions ':=' are in fact not definitions, then you'll have to explain conditioning on an event of probability zero to me please.
Wiki just says for 'conditional cumulative distribution function' or conditional CDF of $X$ and $Y$ on $Z=z$ is $F_{(X,Y)|Z=z}(x,y):=P(X \le x, Y \le y|Z=z)$, but it doesn't quite define $P(X \le x, Y \le y|Z=z)$.
For just 1 continuous random variable conditioned on 1 continuous random variable, it's $P(X \le x, |Z=z) := \int_{-\infty}^{x} f_{X|Z=z}(t) dt$, where $f_{X|Z=z}(t) := \frac{f_{(X,Z)}(x,z)}{f_{Z}(z)}$.
For 2 continuous random variables conditioned on 1 continuous random variable, I think it's $P(X \le x, Y \le y|Z=z) := \int_{-\infty}^{x} \int_{-\infty}^{y} f_{(X,Y)|Z=z}(t,u) du dt$, but then...
What's $f_{(X,Y)|Z=z}(t,u)$? (I guess we do the elementary probability way of thinking: define the pdf before the cdf...) According to this site (see problems 1 and 16), it's $f_{(X,Y)|Z=z}(t,u): = \frac{f_{(X,Y,Z)}(t,u,z)}{f_Z(z)}$. So, I guess I'm right about joint cdf/pdf stuff. I'm just hoping for a reference please.
