# conditional distribution of continuous random variable -- is $P(X<u | Y=y) >0$ an abuse of notation?

Given two random variables, both continuous, $$X$$ and $$Y$$, we know that $$P(X=x)=P(Y=y)=0$$. Is it an abuse of notation or is it correct to have the following?

$$P(X 0$$

In particular, can we claim the following?

$$P(X

We have \begin{align} F_X(u) = P(X< u) &= \int_{y=-\infty}^{\infty} \quad \int_{x=-\infty}^u f(x,y) dx \quad dy \\ &= \int_{y=-\infty}^{\infty} \quad \frac{ P( X < u, y < Y < y+dy)}{dy} \quad dy \\ &= \int_{y=-\infty}^{\infty} \quad { P( X < u| y < Y < y+dy)} f_Y(y)dy{} \\ \end{align}

In the expression above, it looks like we can claim the following

$$P(X

however, in the left hand side we do not have $$dy$$ and in the right hand side we do?

$$P(X

Is the best to say

$$P(X

or

$$P(X

?

Here is a related question:

How to formalize "conditional random variables"

• Are they discrete or continuous random variables? It looks like you are taking them to be discrete so that $P(\{X=x\})$ and $P(\{Y=y\})$ are non-zero. Commented Jul 30, 2020 at 18:17
• all continuous ... updating accordingly Commented Jul 30, 2020 at 18:17
• It has to do with a notion of conditional expectation. We define then conditional probability in a way $\mathbb P(X \in A | \mathcal G) = \mathbb E[1_A | \mathcal G]$ (it's a random variable), where $\mathcal G$ is some $\sigma-$field. Now it can be shown that for $\mathbb E[X|Y] := \mathbb E[X|\sigma(Y)]$ there exists $\phi$ borel such thath $\mathbb E[X|Y] = \phi(Y)$. Then, you would get some $\phi$ such that $\mathbb P(X \le u | Y) = \phi(Y)$. And one can then understand $" \mathbb P(X \le u | Y=y)"$ to be the value $\phi(y)$ Commented Jul 30, 2020 at 18:18
• @DanielS. Are you familiar with basic notions from probability theory such as sample spaces and $\sigma$-algebras? Commented Jul 30, 2020 at 18:23
• In this case try the following statlect.com/fundamentals-of-probability/… . It is fairly simple and provides you with a remedy for defining $P(A|B)$ when $P(B)=0$. Commented Jul 30, 2020 at 18:28