Linearity of expectation

Idea: Count with small questions

Write a count as a sum of yes-or-no indicators, then add their means.

• An indicator's mean is its probability
• Dependence between them does not matter

Formula: Linearity of expectation

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

Holds for any X and Y with a finite mean, independent or not. Also E[aX + b] = aE[X] + b.

Example: Red balls in 3 draws

An urn has $6$ red and $4$ blue. Draw $3$ without replacement. How many red do you expect?

1. Each draw alone is red with chance $\frac{6}{10}$
2. Add the indicators: $3\times \frac{6}{10}=\frac{9}{5}$

Idea: The long way agrees

Listing all hands takes much longer.

• $\left(\genfrac{}{}{0ex}{}{10}{3}\right)=120$ hands
• Weighted count: $\frac{36+120+60}{120}=\frac{9}{5}$

Idea: Pick what the indicator is on

Attach indicators to whatever makes one probability easy.

• A position, a person, a pair of neighbours, a value
• Recipe: an indicator, one probability, then add

Quick check: Idle traders

$4$ orders each go at random to one of $3$ traders. How many traders do you expect to get no order?

Common mistake: Linearity for products

Linearity covers sums only, not products or maxima.

Instead Both draws red: $\frac{6}{10}\times \frac{5}{9}=\frac{1}{3}$, not $\frac{6}{10}\times \frac{6}{10}=\frac{9}{25}$.

Idea: A bet is an indicator

Win $10$ with $\frac{1}{4}$, else lose $2$: a sure $-2$ plus $12$ times a win indicator.

• Mean: $-2+12\times \frac{1}{4}=1$
• A book of positions adds the same way; risk does not

Idea: Say it out loud

Define the indicator in a full sentence: it is $1$ if this happens to item k.

• Say why one item's probability is the same for all
• Say that dependence does not matter here