The lottery question

If you play the lottery every week for a year, how much can you expect to lose?

What do you think?

A game costs $2 to play and pays $1000 with probability 1/1000. What's the expected value of playing 52 weeks?

The answer is simple: we just multiply. That's linearity of expectation.

Expected value: a quick review

Expected Value

For a discrete random variable $X$ with PMF $p(x)$ : $E[X] = \Large\sum_x x \cdot P(X = x)$ It's the probability-weighted average — the "center of mass" of the distribution.

For a fair die: $E[X] = 1 \cdot \Large\frac{1}{6} + 2 \cdot \Large\frac{1}{6} + \cdots + 6 \cdot \Large\frac{1}{6} = 3.5$

You can never actually roll a 3.5, but over many rolls, your average will converge to it.

The linearity property

Linearity of Expectation

For any random variables $X$ and $Y$ (independent or not!) and constants $a$ , $b$ : $E[aX + bY] = a \cdot E[X] + b \cdot E[Y]$

No independence required. This works even when $X$ and $Y$ are heavily dependent.

Try different distributions and scaling constants:

Linearity Explorer

X distribution

E=3.50

Y distribution

E=0.50

a (coefficient of X): 1

-35

b (coefficient of Y): 1

-35

a·E[X]

3.500

b·E[Y]

0.500

E[aX + bY]

4.000

1·(3.50) + 1·(0.50) = 4.000

Why is this surprising?

For most operations, dependence matters. The variance of a sum depends on whether $X$ and $Y$ are correlated. But expectation doesn't care.

Consider: $X$ = temperature in New York tomorrow, $Y$ = number of umbrellas sold tomorrow. They're clearly dependent! Yet:

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

The formula holds without conditions.

What do you think?

100 people each flip a fair coin. What's the expected total number of Heads?

Enter a whole number

See it in action

Run a simulation: X = coin flip (0/1), Y = fair die roll (1–6). Theory predicts E[X] = 0.5, E[Y] = 3.5, E[X+Y] = 4.0. Watch the averages converge:

Linearity Simulation

Samples

avg(X)

0.000

avg(Y)

0.000

avg(X+Y)

0.000

Scaling and shifting

Linearity includes two sub-rules:

Rule	Formula	Example
Scaling	$E[aX] = a \cdot E[X]$	Double all outcomes → double the mean
Shifting	$E[X + c] = E[X] + c$	Add 5 to every outcome → add 5 to the mean
Combined	$E[aX + b] = aE[X] + b$	Converting Celsius to Fahrenheit

A restaurant tip calculation

Your bill is a random variable $B$ with $E[B] = 40$ . You tip 20%, so your total is $T = 1.20 B$ .

What is E[T] = E[1.20B]? (whole number)

Your friend's bill has E[B₂] = 35. What's the expected combined total (with 20% tip each)? (whole number)

A preview: this extends to N variables

Linearity generalizes to any number of random variables:

$\Large E\left[\sum_{i=1}^{n} X_i\right] = \sum_{i=1}^{n} E[X_i]$

We'll use this for indicator variable problems in the next lesson.

Summary

Concept	Key Formula
Expected value	$E[X] = \sum_x x \cdot P(X = x)$
Linearity (sum)	$E[X + Y] = E[X] + E[Y]$
Linearity (scaled)	$E[aX + b] = aE[X] + b$
Independence?	Not needed for linearity!

When you see "expected value of a sum," immediately think linearity. Don't try to find the distribution of the sum — just add the expectations.

Test your understanding

Roll 3 fair dice. What's E[sum]? (decimal, e.g. 0.42)

A company has 200 employees. Each has a 2% chance of calling in sick. Expected sick calls? (whole number)

X and Y are dependent with E[X]=7, E[Y]=3. What is E[X+Y]? (whole number)

What's next

We'll use linearity with indicator variables, a technique that turns hard counting problems into simple sums.