Maybe it's a silly question, but I am not quite sure how to calculate a multivariate distribution. Here is what I have:
Given are $X,Y$ two i.i.d. uniform distributed $U(0,1)$ random variables and $T:=XY$, $Z:=YT$
First I needed to calculate the distributions for $X,Y$ and $T$. I have evaluated them to be $$F_{X,Y}=F_X F_Y=\int\int 1_{[0,1]}(x)1_{[0,1]}(y)dxdy=xy$$ $$F_T=\int P(Y\leq \frac{t}{x})f_X(x)dx=\int^t_0 dx+\int^1_t \frac{t}{x}dx=t-t\cdot log(t)$$
But now I need to calculate the multivariate distributions of $X,T$ and $Z,T$.
Now my question boils down to:
Is $T$ dependent or independent of $X$? How about dependence between $Z$ and $T$? I mean, I at least think that they are obviously dependent, but then how do I calculate the multivariate distributions if I cant use $F_{X,T}\neq F_X F_T$? I don't see how to apply the more general statement $F_{X,T}=F_{X|T}F_T$ to this problem.
Can anyone help clear my mind?