- Given random variables $X, Y, Z$, is joint distribution of $X$ and "$Y|Z$" same as distribution of $(X, Y)|Z$?
- Given random variables $X_1, X_2, Y_1, Y_2$, is joint distribution of "$X_1 | X_2$" and "$Y_1|Y_2$" same as distribution of $(X_1, Y_1) | (X_2, Y_2)$?
- For two independent random vectors $X$ and $Y$, and any two subvectors $X_1$ and $X_2$ of $X$ and any two subvectors $Y_1$ and $Y_2$ of $Y$, will the conditional random vectors "$X_1|X_2$" and "$Y_1|Y_2$" also be independent?
Why? Thanks and regards!