Sample distribution theory using Coarea Formula

Let (Omega, Sigma, p) be a probability measure space and let X : Omega -> R-k be a (vector valued) random variable. We suppose that the probability p(x) induced by X is absolutely continuous with respect to the Lebesgue measure on R-k and set f(x) as its density function. Let phi : R-k -> R-n be a C-1-map and let us consider the new random variable Y = phi(X) : Omega -> R-n. Setting m := max{rank (J phi(x)) : x is an element of R-k}, we prove that the probability p(y) induced by Y has a density function f(y) with respect to the Hausdorff measure H-m on phi(R-k) which satisfiesf(Y)(y) = integral(phi-1(y))f(x)(x)1/J(m)phi(X) dH(k-m)(x), for H-m - a.e. y is an element of phi(R-k).Here J(m)phi is the m-dimensional Jacobian of phi. When J phi has maximum rank we allow the map phi to be only locally Lipschitz. We also consider the case of X having probability concentrated on some m-dimensional sub-manifold E subset of R-k and provide, besides, several examples including algebra of random variables, order statistics, degenerate normal distributions, Chi-squared and "Student's t" distributions.