the change-of-variables (Jacobian) formula
/ Jacobian: juh-KOH-bee-un /
For a smooth one-to-one transformation Y = g(X), there is a shortcut that gives the density of Y in one line, without re-deriving the cdf each time. It is the change-of-variables formula, and it is the cdf method done once and remembered.
Here is why a correction factor appears. A small interval of width dx around x maps to a small interval around y = g(x), but the new interval is stretched or squeezed by the slope of g. The amount of probability in that little interval cannot change — it is the same outcomes relabelled — so if g spreads the interval out, the density must drop to keep the area the same. The formula encodes this: f_Y(y) = f_X(x) times the absolute value of dx/dy, evaluated at x = g^{-1}(y). The factor |dx/dy| (equivalently 1 divided by |g'(x)|) is the one-dimensional Jacobian. Always take the absolute value, because density is never negative even when g is decreasing.
Read it as conservation of probability under relabelling: density times width is preserved, so a steeper g (which stretches widths) thins the density and a flatter g concentrates it. The formula needs g to be monotone (one-to-one) and differentiable on the range of X. If g folds the line, like squaring, split into monotone pieces and add their contributions — or just use the cdf method.
Let X be exponential with rate 1 and Y = X^3 (monotone increasing on x > 0). Then x = y^(1/3) and dx/dy = (1/3) y^(-2/3). So f_Y(y) = f_X(y^(1/3)) times (1/3) y^(-2/3) = e^(-y^(1/3)) times (1/3) y^(-2/3) for y > 0. The Jacobian factor (1/3) y^(-2/3) is what rescales the density to its new shape.
The |dx/dy| factor rescales the old density so total probability is preserved.
This formula is for densities and requires a one-to-one differentiable g. It does NOT apply to discrete variables (use the pmf directly) nor to non-monotone g without splitting into branches. Always take the absolute value of the Jacobian.