CDF Method Deriving Distribution for Transformation

Method

• For Continuous Injective function $h$ on domain $A=(-\infty ,y],y\in \mathbb{R}$
• The inverse image $h^{-1}$ can use CDF of $X$ to find CDF of $Y$ $F_Y(y)=P(X\le h^{-1}(y))=F_X(h^{-1}(y))$

Strictly Decreasing Function Method

$F_Y(y)=P(Y\ge y)=P(h(x)\le y)=P(X\ge h^{-1}(y))=1-F_X(h^{-1}(y))$

Intuition

• You are essentially graphing $x$ and $y$ based off $h(x)$