Definition

$Y = g (X)$

A new Random Variable $Y$ can be defined by applying a function $g$ to the outcomes of a Random Variable $X$

Discrete

Suppose $g : X \to Y$ is Bijective function $P_{Y} (y) = P (Y = y) = p (g (X) = y) = P (X = g^{- 1} (y)) = P_{X} (g^{- 1} (y))$

Let $f_{X} (x)$ be the PDF of Random Variable $X$ with a support $S_{X}$ , $g : X \to Y$ be a differentiable Bijective function, and $x = g^{- 1} (y)$

$f_{Y} (y) = f_{X} (g^{- 1} (y)) \frac{d x}{d y}, y \in S_{Y}$ where $S_{Y} = {y = g (x) ∣ x \in S_{X}}$

If $g$ is increasing, then $F_{Y} (y) = P (Y \leq y) = P (g (X) \leq y) = P (X \leq g^{- 1} (y)) = F_{X} (g^{- 1} (y))$

If $g$ is decreasing, then $F_{Y} (y) = P (Y \leq y) = P (g (X) \leq y) = P (X \geq g^{- 1} (y)) = 1 - F_{X} (g^{- 1} (y))$