Transformations

🎲 Transforming Random Variables: The Magic Shape-Shifter!

The Big Picture

Imagine you have a magical clay that can change shape! You start with a blob (your original random variable), and when you squeeze, stretch, or twist it (transform it), you get a new shape (a new random variable). But here’s the cool part: we can predict exactly what the new shape will look like!

🔄 Random Variable Transforms

What Is a Transform?

Think of a transform like a magic machine. You put something in, and something different comes out!

The Vending Machine Analogy:

• You put in a coin (input X)
• The machine does something (the function g)
• Out pops a snack (output Y = g(X))

Y = g(X)

Input X → [Magic Machine g] → Output Y

Simple Examples

Example 1: Doubling Machine

• If X = your height in feet
• Y = 2X = your height measured in half-feet
• You didn’t grow taller—just measured differently!

Example 2: Squaring Machine

• If X can be -2, -1, 0, 1, or 2
• Y = X² gives us 4, 1, 0, 1, or 4
• Notice: Different inputs can give the SAME output!

graph TD
    A["X = Random Variable"] --> B["Apply Function g"]
    B --> C["Y = g of X"]
    C --> D["New Random Variable!"]

Why Does This Matter?

Every time you:

• Convert temperatures (Celsius to Fahrenheit)
• Calculate areas from lengths
• Compute profits from sales

You’re transforming random variables!

📊 PDF of Transformed Variable

The Detective Work

When you transform X into Y, the probabilities must go somewhere! Finding the new PDF is like being a detective—tracking where all the probability “went.”

The Golden Formula

For monotonic (always increasing or always decreasing) functions:

$f_Y(y) = f_X(x) \cdot \left| \frac{dx}{dy} \right|$

In simple words:

1. Take the original PDF
2. Multiply by “how stretched” the transformation is
3. That’s your new PDF!

The Stretching Analogy

Imagine probability is like paint on a rubber band:

• Stretch the band → paint spreads thinner
• Squeeze the band → paint gets thicker

The $|dx/dy|$ tells us the stretching factor!

Step-by-Step Example

Problem: If X is uniform on [0, 1], find PDF of Y = X²

Step 1: Write the relationship

• Y = X², so X = √Y

Step 2: Find the derivative

• dx/dy = 1/(2√y)

Step 3: Apply the formula $f_Y(y) = f_X(\sqrt{y}) \cdot \frac{1}{2\sqrt{y}} = 1 \cdot \frac{1}{2\sqrt{y}} = \frac{1}{2\sqrt{y}}$

Result: Valid for 0 < y < 1

graph TD
    A["Original PDF of X"] --> B["Find inverse: x = g⁻¹ of y"]
    B --> C["Calculate dx/dy"]
    C --> D["Multiply: fX times abs dx/dy"]
    D --> E["New PDF of Y!"]

Non-Monotonic Functions

What if the function isn’t always increasing or decreasing?

Example: Y = X² when X can be negative OR positive

Both X = 2 and X = -2 give Y = 4!

Solution: Add up contributions from each “branch”:

$f_Y(y) = \sum_{\text{all } x_i} f_X(x_i) \cdot \left| \frac{dx}{dy} \right|_{x=x_i}$

➕ Sum of Random Variables

The Party Problem

You’re throwing a party!

• Guest A brings X cookies
• Guest B brings Y cookies
• Total cookies = X + Y

How do we find the probability of having exactly 10 cookies?

The Convolution Magic

The PDF of Z = X + Y is found by convolution:

$f_Z(z) = \int_{-\infty}^{\infty} f_X(x) \cdot f_Y(z - x) , dx$

What Is Convolution?

Think of it as “sliding and multiplying”:

1. Flip one PDF horizontally
2. Slide it across the other
3. At each position, multiply and add up
4. That gives you one point of the new PDF

graph TD
    A["PDF of X"] --> C["Convolution"]
    B["PDF of Y"] --> C
    C --> D["PDF of Z = X + Y"]
    D --> E["Slide, Multiply, Sum!"]

Visual Example: Two Dice

Rolling two dice (each uniform 1-6):

The result? A triangular distribution peaking at 7!

Key Properties

1. Order doesn’t matter: X + Y = Y + X (same distribution)
2. Means add: E[X + Y] = E[X] + E[Y]
3. Variances add (if independent): Var(X + Y) = Var(X) + Var(Y)

🔔 Sum of Normal Random Variables

The Beautiful Miracle

Here’s something amazing: Normal plus Normal equals Normal!

This is like mixing blue paint with blue paint—you still get blue paint!

The Magic Formula

If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, then:

$Z = X + Y \sim N(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2)$

In plain English:

• Means add up
• Variances add up
• It’s still normal!

Real-World Example: The Height Problem

Setup:

• Your height X ~ N(170 cm, 25 cm²)
• Your friend’s height Y ~ N(165 cm, 16 cm²)
• Combined height Z = X + Y

Result: $Z \sim N(170 + 165, 25 + 16) = N(335 \text{ cm}, 41 \text{ cm}^2)$

Standard deviation: √41 ≈ 6.4 cm

graph TD
    A["X ~ Normal μ₁, σ₁²"] --> C["Add Them"]
    B["Y ~ Normal μ₂, σ₂²"] --> C
    C --> D["Z ~ Normal μ₁+μ₂, σ₁²+σ₂²"]
    D --> E["Still a Bell Curve!"]

Why This Is Amazing

For other distributions:

• Uniform + Uniform = Triangular (not uniform!)
• Exponential + Exponential = Gamma (not exponential!)

But for Normal:

• Normal + Normal = Normal ✨

This is called stability—the normal distribution is “stable” under addition!

Extension: Many Normal Variables

Adding n independent normal variables:

$\sum_{i=1}^{n} X_i \sim N\left(\sum \mu_i, \sum \sigma_i^2\right)$

Special case: If all X_i are identical N(μ, σ²):

$\sum_{i=1}^{n} X_i \sim N(n\mu, n\sigma^2)$

The Sample Mean Connection

If you take n samples from N(μ, σ²) and average them:

$\bar{X} = \frac{1}{n}\sum X_i \sim N\left(\mu, \frac{\sigma^2}{n}\right)$

The mean stays the same, but variance shrinks by factor n!

This is why averaging reduces noise!

🎯 Quick Summary

🚀 You Did It!

You now understand:

• ✅ How to transform random variables
• ✅ How to find the new PDF after transformation
• ✅ How to add random variables (convolution)
• ✅ The beautiful property of normal distributions

The magic insight: Probability never disappears—it just moves around! When you transform or add variables, you’re reshuffling the probability, and now you know exactly how to track it!