Math Atom · M-C02

Expectation

8 minMath AtomProbability

Know these first

By the end you will be able to

  1. 1Calculate the expected value of a discrete random variable as a probability-weighted average.
  2. 2Check that a set of probabilities is valid: each between 0 and 1, adding up to 1.
  3. 3Use linearity: E[aX + b] = aE[X] + b and E[X + Y] = E[X] + E[Y].
  4. 4Find the expected payoff of a simple bet or option.

Why you need it

A lender does not know which borrowers will default, and an option buyer does not know where the share price will end up. But both can ask: on average, what do I expect to get? That number is the expected value.

Expected values also sit behind expected credit losses, insurance premiums and average portfolio returns, which later levels build on.

The idea

The long-run average

Roll a die once and you get a whole number from 1 to 6. You cannot predict it. But roll it hundreds of times and the average of all your rolls settles down to a predictable number. Try it.

Roll a die, many times

0 rolls
123456expected value 3.5number of rolls →

Running average –

1: 0 · 2: 0 · 3: 0 · 4: 0 · 5: 0 · 6: 0

The number the average settles on is the expected value. For a fair die it is 3.5: a number you can never actually roll. The expected value is not a prediction of the next result; it is the centre of gravity of all the possible results, each weighted by how likely it is.

Picture the outcomes as weights placed along a ruler, each weight as heavy as its probability. The expected value is the point where the ruler balances.

The rules

A quantity that depends on chance is called a random variable. Here is how to find its expected value.

Linearity. Expected values behave very nicely when you scale, shift or add random variables.

The rules, for reference

E[X]=∑i=1npi xi,0≤pi≤1,∑i=1npi=1\mathbb{E}[X] = \sum_{i=1}^{n} p_i\,x_i, \qquad 0 \le p_i \le 1, \qquad \sum_{i=1}^{n} p_i = 1
E[aX+b]=a E[X]+bE[X+Y]=E[X]+E[Y]\mathbb{E}[aX+b] = a\,\mathbb{E}[X] + b \qquad \mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]

For a bent function such as max: apply it to each outcome first, then average. In general E[f(X)]≠f(E[X])\mathbb{E}[f(X)] \neq f(\mathbb{E}[X]).

Try it

Type outcomes and probabilities, or load an example. Watch the balance point move as you shift probability from one outcome to another. Then try probabilities that do not add up to 1, and use a and b to check linearity.

Try itExpectation calculator

Open the probability calculator →
Load:
Outcome xProbability p (0.3 or 1/6)p × x
15
−14

Expected value E[X]

1

Sum of probabilities1
Plain average of x15
Bars show each outcome's probability. The triangle is the balance point, E[X].
50−20E[X] = 1

Linearity: transform every outcome to a·x + b and average again.

E[1X+0]=1=1×1+0\mathbb{E}[1X + 0] = 1 = 1 \times 1 + 0

Worked examples

1. Is this bet worth taking?

Step 1 of 4
  1. 1

    The set-up

    A bet pays €50 with probability 0.3 and loses €20 with probability 0.7.

2. What a lender expects back

Step 1 of 3
  1. 1

    The set-up

    A bank in Hong Kong lends HK$1,000 for a year (ignore interest). With probability 98% it is repaid in full; with probability 2% the borrower defaults and the bank recovers HK$400.

3. The expected payoff of an option

Step 1 of 4
  1. 1

    The set-up

    A share will be worth 80, 100, 130 in a year, with probabilities 0.25, 0.5, 0.25. An option pays max⁡(S−100, 0)\max(S - 100,\, 0) (see M-A09 for max).

Common mistakes

Each line below is a tempting slip. Spot what is wrong, then check.

Reaction

Check what stuck

Question 1 · objective 1

What is the expected score when you roll a fair six-sided die?

Question 2 · objective 4

A bet wins 30 with probability 0.25 and loses 10 with probability 0.75. What is its expected gain?

Question 3 · objective 3

E[X] = 4. What is E[3X + 2]?

Question 4 · objective 4

A share will be worth 40 or 60 in a year, each with probability 0.5. What is the expected payoff of max(S − 50, 0)?

Question 5 · objective 2

Which list could be the probabilities of all the outcomes of a random variable?

Glossary

Random variable
A number whose value depends on chance, such as the score on a die or the payoff of a bet. Written with a capital letter, X.
Outcome
One of the values a random variable can take.
Probability
A number from 0 (impossible) to 1 (certain) saying how likely an outcome is. The probabilities of all the outcomes add up to 1.
Expected value
The probability-weighted average of a random variable's outcomes, written E[X]: the long-run average over many repeats.
Linearity
E[aX + b] = aE[X] + b and E[X + Y] = E[X] + E[Y]: scaling, shifting and adding pass straight through an expectation.
Fair bet
A bet whose expected gain is zero.