3

Transformations of Random Variables

Method of CDFs

The Big Picture

Given a random variable $X$ with a known distribution, we often need to find the distribution of a transformed random variable $Y = g(X)$.

Key Insight

The Method of CDFs is the most fundamental approach: find $F_Y(y)$ first, then differentiate to get $f_Y(y)$.

$f_Y(y) = \frac{d}{dy} F_Y(y)$

The 6-Step Method

Worked Example: $Y = \ln X$

Problem Statement

Let $X \sim \text{Expo}(1)$. Find the PDF of $Y = \ln X$.

Step 1 / 5

Interactive Visualization

Original: $X \sim \text{Expo}(1)$

Transformed: $Y = \ln X$

Animate Transformation

Watch how sample points from $X$ are transformed through $Y = \ln X$

Monotonicity Quick Reference

Increasing $g(x)$

$g(X) \le y \iff X \le g^{-1}(y)$
$F_Y(y) = F_X(g^{-1}(y))$

Examples: $Y = 2X+3$, $Y = e^X$, $Y = \ln X$ (for $X > 0$)

Decreasing $g(x)$

$g(X) \le y \iff X \ge g^{-1}(y)$
$F_Y(y) = 1 - F_X(g^{-1}(y))$

Examples: $Y = -X$, $Y = 1/X$ (for $X > 0$)

Non-Monotonic Functions

For functions like $Y = X^2$ where $X$ can be negative, you must split the event:

$P(X^2 \le y) = P(-\sqrt{y} \le X \le \sqrt{y})$

Quick Check

Key Takeaways

1️⃣

Start with CDF

Always write $F_Y(y) = P(Y \le y)$ first

2️⃣

Check Monotonicity

Increasing vs decreasing changes the inequality direction

3️⃣

Don't Forget Chain Rule

The Jacobian factor adjusts for the "stretching" of space