General principle

• If the transformation, \(g\) is a one-to-one mapping, then:
  • \(F_Y(y) = \mathbb{P}[Y\leq Y] = \mathbb{P}[g(X)\leq Y] = \mathbb{P}[X\leq g^{-1}(y)] = F_X(g^{-1}(y))\).
  • Via the chain rule: \(f_Y(y) = \left( \dfrac{d g^{-1}(y)}{dy} \right) f_X(g^{-1}(y))\).