Sample distribution theory using Coarea Formula

Let (Ω, Σ, p) be a probability measure space and let X: Ω→Rk be a (vector valued) random variable. We suppose that the probability pX induced by X is absolutely continuous with respect to the Lebesgue measure on Rk and set fX as its density function. Let ϕ: Rk→Rn be a C1-map and let us consider the new random variable Y=ϕ (X): Ω→Rn. Setting m: =maxrank (Jϕ (x) ): x∈Rk, we prove that the probability pY induced by Y has a density function fY with respect to the Hausdorff measure Hm on ϕ (Rk) which satisfies fY (y) =∫ϕ−1 (y) fX (x) 1Jmϕ (x) dHk−m (x), for Hm−a. e. y∈ϕ (Rk). Here Jmϕ is the m-dimensional Jacobian of ϕ. When Jϕ has maximum rank we allow the map ϕ to be only locally Lipschitz. We also consider the case of X having probability concentrated on some m-dimensional sub-manifold E⊆Rk and provide, besides, several examples including algebra of random variables, order statistics, degenerate normal distributions, Chi-squared and “Student's t” distributions.