Keeping Probabilities Honest: The Jacobian Adjustment

The Jacobian adjustment is a necessary scaling factor when transforming random variables to ensure the total probability remains 1. Using an analogy of sand on a flexible sheet, a transformation that compresses the axis increases probability density, while stretching it decreases density. The mathematical derivation shows that the new probability density function, f_Y(y), is the original density, f_X(x), multiplied by the absolute value of the derivative of the inverse transformation, |dx/dy|. Empirical simulations using Python confirm that the histogram of transformed data aligns with the theoretically correct, Jacobian-adjusted distribution, not the naive, unadjusted one.