In Degroot: Probability and Statistics, there's a general formula for functions of Random Variables.
Suppose: X1, X2... Xn have a joint continuous p.d.f f(.), and there're one-to-one correspondence functions Y1 = r1(X1, ... Xn), Y2 = r2(X1, ... Xn), ... Yn = rn(X1, ... Xn), as well as the inverse X1 = s1(Y1,...Yn), Y2 = s2(Y1,...Yn),...,Yn = sn(Y1,...Yn). Assume si(.) has partial derivatives, so there's Jacobian determinant |J| of si(.).
Assume vector X is not 0 in area D, and the correspondence area is T for vector Y. Then, the joint p.d.f of Y is
g(y1, y2,...,yn) = f(s1, s2, ... sn) * |J| for(y1, y2,...,yn) in T; 0, otherwise.
The author didn't give derivation of this formula. How to derive it rirugously? (one can use change of variable formula in multi-calculus directly without proof)
