Distribution of a function of a random variable

Definition:

• Let $X$ be a Continuous random variable having PDF $f_X$.
• Suppose that $g(x)$ is a strictly monotonic, differentiable function of $x$.
• The random variable $Y$ defined by $Y=g(X)$ has a PDF given by:
  • $f_Y=\{\begin{array}{llc}{f}_X[g^{-1}(y)]& |\frac{d}{dy}{g}^{-1}(y)|& \text{if }y=g(x)\text{ for some }x\\ 0& & \text{if }y\ne g(x)\text{ for all }x\end{array}$
    • meaning $f_Y=f_X[g^{-1}(y)]\times \frac{d}{dy}{g}^{-1}y$
    • Example:
      • $Y=X^3\to P(Y<y)=P(X^3<y)=P(X<\sqrt[3]{y})=F_X(\sqrt[3]{y})$
      • Differentiate that, we have PDF of $f_X$ in terms of $y$
  • where $g^{-1}(y)$ is defined to be equal to that value of $x$ such that $g(x)=y$

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