I'm working through a course in Probability (2nd/3rd year) and would like to clarify some idea on joint distributions.
Suppose for example we have independent random variables $(Z_1,Z_2)$ from a distribution, which we will take to be standard normal, i.e. $N[0,1]$, and then we define variables $X(Z_1, Z_2)$ and $Y(Z_1, Z_2)$, functions of $Z_1,Z_2$, how do we find the marginal distributions of $X$ and $Y$ and their joint distribution?
If we look at a specific case, say for example $X=Z_1+Z_2$ and $Y=2Z_1+Z_2$, how could we find the conditional expectation of $Y$ say if we fix a value for $X$, so say $E[Y|X=\alpha]$ for some $\alpha >0$?
I would be very appreciative of anyone who could help clarify these ideas. Best, MM.