Linear Change of Variables for Densities Many problems require finding the distribution of some function of a continuous RV X, say Y = g(X), from the distribution of X. Let’s start with the case where Y is a linear function of X, Y = aX + b.

One-to-One Change of Variables for Densities Let X be a RV with density fX(x) on the range (a,b). Let Y = g(X) where g is either strictly increasing or strictly decreasing on (a,b). The range of Y is then an interval with endpoints g(a) and g(b). The density of Y on this interval is The equation y = g(x) must be solved for x in terms of y, and this value of x substituted into fX(x) and dy/dx. This will leave an expression for fY(y) entirely in terms of y.