Functions of Random Variables

Compact study note.

Summary

Applying one measurable function to a random variable creates another random variable. Its distribution is found by pushing probability through the transformation.^[1]

Prerequisites

• Random Variable
• Distribution Induced by a Random Variable

Notation and Assumptions

If $Y=g(X)$ and $g$ is Borel-measurable, then $Y$ is a random variable and $P(Y\in B)=P(X\in g^{-1}(B))$ .

Essential Result

For an injective differentiable transform $Y=g(X)$ with inverse $x=g^{-1}(y)$ , $f_Y(y)=f_X(x)|dx/dy|$ on the transformed support.

Small Example

If $X\sim \mathrm{Uniform}(0,1)$ and $Y=X^2$ , then $F_Y(y)=P(X\le \sqrt{y})=\sqrt{y}$ for $0\le y\le 1$ .

Common Mistakes

• Forgetting to transform the support.
• Applying the derivative formula to non-injective transformations without splitting branches.

Connections

• Cumulative Distribution Function
• Probability Density Function
• Log-Normal Distribution

References