Let $X_1$ and $X_2$ be two real random variables such that $X_1 \sim \mathbb{Q_1}$ and $X_2 \sim \mathbb{Q_2}$.
We assume that $\rho = Corr(X_1, X_2) \neq 0$.
What can we say about the conditional distribution of $X_2$ given $X_1$ ?
What I would like to know is : Let $\omega \in \Omega$ the associated sample space, if we suppose $X_1(\omega)$ is an outlying value, then what can we say about the value taken by $X_2(\omega)$ since they are correlated ?
For instance: is it true that $\mathbb{P}(X_2 \in V_2 | X_1 = \mathbb{E}[X_1]) < \mathbb{P}(X_2 \in V_2 | X_1 = v_1)$ where $V_i$ is such that $\mathbb{P}(X_i \in V_i) = \alpha$ (outlying sets; one can think $\alpha$ small) and $v_1 \in V_1$.
In other words, given two correlated random variables, what can we say about the probability of observing an outlier from the second variable when we know the first one has an outlying value ?
I am sorry if it is not clear enough? I just want to know if when twos random experiences are correlated, the production of outliers is correlated, and if it is true how can we show it mathematically ?