Change of Variable

The change of variable formula is:

\[\int^{a}_{b} f(x) dx = \int^{y(b)}_{y(a)} f(g(y)) \frac{dg}{dy} dy\]

where \(x = g(y)\).

Suppose \(X\) is a probability distribution with density \(p_X(x)\) and \(Y = g(X) \implies X = g^{-1}(Y)\). For the probability of \(Y\) we have \(P(g(a) \leq Y \leq g(b)) = P(a \leq X \leq b)\). Then,

\[P(g(a) \leq Y \leq g(b)) = P(a \leq X \leq b) = \int^{a}_{b} p_X(x) dx = \int^{g(b)}_{g(a)} p_X(g^{-1}(Y))|\frac{dg}{dy}| dy\]

That is:

\[p_Y(y) = p_X(g^{-1}(Y))|\frac{dg}{dy}|\]

Reference

https://www.cl.cam.ac.uk/teaching/0708/Probabilty/prob11.pdf