It is clear that H is a bijective mapping, and the Jacobian of H − − 1 is given by: d H − − 1 ( y , x ~ ~ ) d y d x ~ ~ = [ 1 − − d V ( x ~ ~ ) d x ~ ~ 0 n × × 1 I n × × n ] , which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that f Y , X ( y , x ) = f X ( x ) δ δ ( y − − V ( x ) ) , which if marginalized over x leads to the desired probability density function.