Expected Value and Variance

🎲 Expected Value & Variance: The Crystal Ball of Probability

Imagine you have a magical crystal ball that can predict the “average outcome” of any game you play. That’s exactly what Expected Value does! And Variance? That tells you how wild and unpredictable the ride might be.

🎯 The Big Picture

Think of a lemonade stand. Some days you sell 10 cups, some days 5, some days 20. The expected value is like asking: “On a typical day, how many cups will I sell?” And variance tells you: “How crazy do my sales swing around?”

1️⃣ Indicator Random Variables: The Yes/No Detectives 🔍

What Are They?

An indicator random variable is the simplest detective in probability. It only asks ONE question:

• Did the thing happen? → Answer: 1 (Yes!)
• Did it NOT happen? → Answer: 0 (Nope!)

That’s it. Just 1 or 0. Like a light switch: ON or OFF.

Simple Example 🍪

You flip a coin. Let’s create an indicator variable I for “getting heads”:

• Heads shows up → I = 1
• Tails shows up → I = 0

Why Are They Useful?

They’re like counting helpers! Want to count how many heads in 10 flips? Just add up all the indicators:

Total Heads = I₁ + I₂ + I₃ + … + I₁₀

Each I is either 0 or 1, so the sum = total count of heads!

Flip 1: Heads → I₁ = 1
Flip 2: Tails → I₂ = 0
Flip 3: Heads → I₃ = 1
...
Sum = 1 + 0 + 1 + ... = Count!

2️⃣ Computing Expected Value: The Magic Average 🎩

The Core Idea

Expected Value (E[X]) = The average outcome if you played the game forever.

It’s NOT what you’ll get next time. It’s what happens “on average, in the long run.”

The Formula

For each possible outcome, multiply: (value) × (probability of that value)

Then add them all up!

E[X] = Σ (value × probability)
     = x₁·P(x₁) + x₂·P(x₂) + ...

Simple Example 🎲

Rolling a fair die. What’s the expected value?

E[X] = 1×(1/6) + 2×(1/6) + 3×(1/6) + 4×(1/6) + 5×(1/6) + 6×(1/6)

E[X] = (1+2+3+4+5+6)/6 = 21/6 = 3.5

🤔 Wait, you can’t roll 3.5! True! Expected value is a theoretical average, not a real outcome.

3️⃣ Linearity of Expectation: The Superpower ⚡

The Magic Rule

This is possibly the MOST POWERFUL tool in probability:

E[X + Y] = E[X] + E[Y]

ALWAYS. No conditions. No exceptions. Even if X and Y are connected!

Why Is This Mind-Blowing?

Normally in math, you can’t just split things apart. But here? You CAN!

Simple Example 🎲🎲

Roll TWO dice. What’s the expected sum?

Instead of listing all 36 combinations…

E[Die1 + Die2] = E[Die1] + E[Die2] = 3.5 + 3.5 = 7

Done! That’s why 7 is the most “expected” sum in dice games!

Another Example 🪙

Flip a coin 100 times. Expected number of heads?

Let I₁, I₂, …, I₁₀₀ be indicators for each flip being heads.

E[Total Heads] = E[I₁] + E[I₂] + … + E[I₁₀₀]

Each E[Iᵢ] = 0.5 (since P(heads) = 0.5)

= 100 × 0.5 = 50 heads expected

4️⃣ Expected Value of an Indicator: Super Simple! 🌟

The Beautiful Truth

For an indicator variable I that equals 1 when event A happens:

E[I] = P(A)

That’s it! The expected value IS just the probability!

Why?

E[I] = 0 × P(not A) + 1 × P(A)
     = 0 + P(A)
     = P(A)

Example 🎯

Indicator for rolling a 6 on a die:

• E[I₆] = P(rolling 6) = 1/6 ≈ 0.167

This means: If you rolled forever, about 16.7% of rolls would be 6s.

5️⃣ Mean of Distribution: The Center of Gravity ⚖️

What Is It?

The mean (μ) IS the expected value! Same thing, different name.

Think of it as the “balance point” of a distribution.

graph TD
    A["Distribution"] --> B["Mean = Expected Value"]
    B --> C["μ = E[X"]]
    C --> D["The Balance Point"]

Why “Balance Point”?

Imagine outcomes on a seesaw. The mean is where you’d put the pivot to balance all the probability weights!

Example: Loaded Die 🎲

A die is loaded: P(6) = 0.5, all others = 0.1 each

μ = E[X] = 1(0.1) + 2(0.1) + 3(0.1) + 4(0.1) + 5(0.1) + 6(0.5)

= 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 3.0 = 4.5

The mean shifts toward 6 because it’s more likely!

6️⃣ Computing Variance: Measuring the Chaos 🌪️

The Core Idea

Variance (Var(X)) measures: How spread out are the outcomes from the mean?

• Low variance = outcomes cluster near the mean (predictable)
• High variance = outcomes scatter wildly (unpredictable)

The Formula

Var(X) = E[(X - μ)²]

Or the easier computation formula:

Var(X) = E[X²] - (E[X])²

This says: “Average of squares minus square of average”

Step-by-Step Example 🎲

Fair die: We know E[X] = 3.5

Step 1: Find E[X²]

E[X²] = 1²(1/6) + 2²(1/6) + 3²(1/6)
      + 4²(1/6) + 5²(1/6) + 6²(1/6)
      = (1+4+9+16+25+36)/6
      = 91/6 ≈ 15.17

Step 2: Apply formula

Var(X) = E[X²] - (E[X])²
       = 91/6 - (3.5)²
       = 15.17 - 12.25
       = 2.92

7️⃣ Variance Properties: The Toolkit 🧰

Property 1: Constants Don’t Vary

Var© = 0 where c is any constant

Makes sense! If it’s always the same number, there’s zero spread!

Property 2: Adding Constants Doesn’t Change Variance

Var(X + c) = Var(X)

Shifting everything left or right doesn’t change the spread!

Property 3: Scaling Squares the Variance

Var(cX) = c² × Var(X)

Double all values? Variance becomes 4× bigger!

Property 4: For INDEPENDENT Variables Only

Var(X + Y) = Var(X) + Var(Y) (only if X, Y are independent)

⚠️ Warning: Unlike linearity of expectation, this ONLY works for independent variables!

Quick Reference Card

8️⃣ Standard Deviation: Making Variance Friendly 📏

The Problem with Variance

Variance is in “squared units.” If X is in dollars, Var(X) is in “dollars squared” – weird!

The Solution

Standard Deviation (σ) = √Var(X)

Now it’s back in the same units as X!

Example 🎲

We found Var(die) ≈ 2.92

σ = √2.92 ≈ 1.71

This means: “Outcomes typically differ from the mean (3.5) by about 1.7”

The Intuition

graph TD
    A["Roll a Die"] --> B["Mean = 3.5"]
    B --> C["Most rolls within 3.5 ± 1.7"]
    C --> D[That's roughly 2 to 5]

Standard deviation gives you a “typical distance from average.”

🎯 Putting It All Together

The Story So Far

graph TD
    A["Random Variable X"] --> B["Expected Value E X"]
    A --> C["Variance Var X"]
    B --> D["The average outcome"]
    C --> E["How spread out"]
    E --> F["Standard Deviation σ"]
    F --> G["Spread in same units"]

Real-Life Example: Test Scores 📝

A class has test scores: Mean = 75, Standard Deviation = 10

• E[Score] = 75 → Average student gets 75
• σ = 10 → Most students score between 65-85
• Var = 100 → The squared spread

The Key Takeaways

1. Indicator variables = Simple 0/1 counters
2. Expected value = Long-run average (Σ value × probability)
3. Linearity = E[X+Y] = E[X] + E[Y] (ALWAYS!)
4. E[indicator] = Just the probability!
5. Mean = Another name for expected value
6. Variance = E[X²] - (E[X])² measures spread
7. Variance properties = Know the rules for constants & scaling
8. Standard deviation = √Variance (back to original units)

💡 Final Wisdom

“Expected value tells you WHERE to aim. Variance tells you how WILD the journey might be.”

Like a GPS: Expected value is your destination. Standard deviation is the traffic unpredictability!

You now have the crystal ball (expected value) AND the chaos meter (variance). Go forth and predict! 🎲✨