The OP wrote:
I know the joint pdf is $24y_2(y_3^2)(y_4^2)$ but ...
Actually, this is incorrect (and has been left uncorrected for over 10 months). The joint pdf is $24y_2(y_3^2)(y_4^3)$.
To see this, let $(X_1, ..., X_4)$ have joint pdf $f(x_1, ..., x_4)$:
Then the joint pdf of random variables $(Y_1, ..., Y_4)$, say $g(y_1, ..., y_4)$, can be derived automatically with:
where I am using the
Transform function from the
mathStatica add-on to Mathematica to do the grunt work for me (I am one of the developers of the former), and with domain of support:
As to independence: $(Y_1, ..., Y_4)$ are said to be mutually stochastically independent if and only if their joint pdf is equal to the product of the marginals. This is easily shown to be the case here: