Conditional Expectation

Conditional Expectation: The Magic of Smart Predictions

The Cookie Jar Story

Imagine you have a magical cookie jar. You can’t see inside, but your wise grandma tells you something special: “When the lid is red, there are usually chocolate cookies. When it’s blue, there are usually vanilla cookies.”

This is conditional expectation — making smarter predictions when you know something extra!

What is Conditional Expectation?

The Simple Idea

Think of it like this:

• Regular expectation = “How many candies do I usually get?”
• Conditional expectation = “How many candies do I get when it’s my birthday?”

The second question uses extra information (it’s your birthday!) to make a better guess.

Real Life Example

🎮 Video Game Scores:

• Your average score = 50 points (regular expectation)
• Your average score when you’ve practiced = 75 points (conditional expectation)
• Your average score when you’re tired = 30 points (conditional expectation)

See? Knowing the condition changes your prediction!

The Formula (Made Simple)

For Two Random Variables X and Y

E[X | Y = y] means: “What’s the average value of X, given that Y equals y?”

E[X | Y = y] = Σ x · P(X = x | Y = y)

In Plain English:

1. Look at all possible values of X
2. Multiply each by its conditional probability (given Y = y)
3. Add them up

Example: The Ice Cream Truck

E[X | Sunny] = 100 × 0.7 + 80 × 0.3 = 94 ice creams

E[X | Rainy] = 20 × 0.6 + 40 × 0.4 = 28 ice creams

Knowing the weather helps predict sales!

Conditional Expectation as a Random Variable

Here’s a mind-bending idea: E[X | Y] itself is a random variable!

Why?

Because Y can take different values. Each value of Y gives a different conditional expectation.

graph TD
    A["Y = ?"] --> B{What value?}
    B --> C["Y = sunny"]
    B --> D["Y = rainy"]
    C --> E["E[X|sunny] = 94"]
    D --> F["E[X|rainy] = 28"]

E[X | Y] = a function that depends on Y’s outcome.

The Magic Property

E[E[X | Y]] = E[X]

This says: “If you average the conditional expectations, you get the regular expectation!”

The Total Expectation Theorem

The Big Idea: Divide and Conquer!

Imagine you want to know the average height of kids in a school:

• Instead of measuring everyone at once…
• Measure each class separately, then combine!

This is the Total Expectation Theorem!

The Formula

E[X] = Σ E[X | Y = y] · P(Y = y)

In words:

1. Find the conditional expectation for each condition
2. Weight each by how likely that condition is
3. Add them all up

Visual Flow

graph TD
    A["Total Expected Value E X"] --> B["Split by Conditions"]
    B --> C["Condition 1: E[X|Y=1] × P#40;Y=1#41;"]
    B --> D["Condition 2: E[X|Y=2] × P#40;Y=2#41;"]
    B --> E["Condition 3: E[X|Y=3] × P#40;Y=3#41;"]
    C --> F["Add All Together"]
    D --> F
    E --> F
    F --> G["= E[X]"]

Example: The School Bus Problem

Setup

A school has two buses:

• Bus A (60% of students): Average travel time = 20 minutes
• Bus B (40% of students): Average travel time = 35 minutes

Question: What’s the average travel time for a random student?

Solution Using Total Expectation

E[Time] = E[Time | Bus A] × P(Bus A)
        + E[Time | Bus B] × P(Bus B)

E[Time] = 20 × 0.6 + 35 × 0.4
E[Time] = 12 + 14
E[Time] = 26 minutes

Example: The Dice Game

Setup

Roll a die. If you get 1-3, flip 1 coin. If you get 4-6, flip 2 coins. X = number of heads.

Find E[X].

Step 1: Conditional Expectations

If die shows 1-3 (one coin):

• E[X | low roll] = 0.5 (one coin, 50% chance of heads)

If die shows 4-6 (two coins):

• E[X | high roll] = 1.0 (two coins, each 50% heads = 1 expected)

Step 2: Apply Total Expectation

E[X] = E[X | low] × P(low) + E[X | high] × P(high)
E[X] = 0.5 × 0.5 + 1.0 × 0.5
E[X] = 0.25 + 0.5
E[X] = 0.75 heads

Why This Matters: Real Applications

1. Insurance Companies

• E[claim | young driver] vs E[claim | experienced driver]
• They charge different premiums based on conditions!

2. Weather Forecasting

• E[temperature | cloudy] vs E[temperature | sunny]
• Forecasters update predictions based on conditions.

3. Stock Market

• E[return | recession] vs E[return | boom]
• Investors adjust strategies based on economic conditions.

Key Properties to Remember

Property 1: Pulling Out Constants

If a is a constant:

E[aX | Y] = a · E[X | Y]

Property 2: Adding Variables

E[X + Z | Y] = E[X | Y] + E[Z | Y]

Property 3: Known Functions

If g(Y) is a function of Y only:

E[g(Y) | Y] = g(Y)

Property 4: The Tower Rule

E[E[X | Y]] = E[X]

This is just the Total Expectation Theorem in disguise!

The Continuous Case

For continuous random variables, sums become integrals:

Conditional Expectation

E[X | Y = y] = ∫ x · f(x|y) dx

Total Expectation

E[X] = ∫ E[X | Y = y] · f(y) dy

Example: Height and Weight

If average weight given height h is: E[Weight | Height = h] = 2h + 30

And height is uniform from 150cm to 190cm:

E[Weight] = ∫ (2h + 30) × (1/40) dh
          = (1/40) × [h² + 30h] from 150 to 190
          = 370 kg (just an example!)

Summary: Your Cheat Sheet

The Final Takeaway

Conditional expectation is like being a detective:

• Without clues, you make general guesses
• With clues (conditions), you make smarter predictions
• The Total Expectation Theorem lets you combine all your clues!

Remember the cookie jar: knowing the lid color helps you predict the cookies inside! 🍪

Now you’re ready to make predictions like a probability master!