Keeping Probabilities Honest: The Jacobian Adjustment

The Jacobian adjustment is a critical scaling factor needed when transforming random variables to ensure probability distributions remain valid (integrate to 1). When transforming X to Y = g(X), naively substituting the inverse into the original PDF produces incorrect probabilities—the adjustment factor |dx/dy| compensates for how transformations stretch or compress different regions of the probability space. The article builds intuition using a "sand on rubber sheet" analogy, derives the mathematical formula step-by-step, demonstrates the error empirically with simulations, and shows a real-world application in histogram equalization for image processing.