Discrete random variables and expected value

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What a discrete random variable is

A random variable $X$ is a quantity whose value depends on chance (e.g. the score on a die).
It is discrete when it can only take separate, countable values (0, 1, 2, …) — usually a count.
A probability distribution lists each value with its probability, often as a table.
The probabilities must add to 1, because some outcome is certain to happen:

$\sum P(X = x) = 1$

Expected value (the mean)

The expected value $E(X)$ is the long-run average value of the variable — what you'd expect on average over many repeats.
Multiply each value by its probability and add:

$E(X) = \sum x \cdot P(X = x)$

The expected value does not have to be one of the possible outcomes.

Worked ExampleFinding an expected value

A game's payout $X$ (in dollars) has the distribution below. Find the expected payout.

$x$	0	1	2	3
$P(X = x)$	0.1	0.3	0.4	0.2

Step 1 — Check the probabilities sum to 1

$0.1 + 0.3 + 0.4 + 0.2 = 1 \;\checkmark$

Step 2 — Multiply each value by its probability and add

$E(X) = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2)$

$= 0 + 0.3 + 0.8 + 0.6 = 1.7$

The expected payout is $1.70.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A random variable $X$ takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Verify the probabilities are valid.

Merit

For the distribution $x = 1, 2, 3$ with $P = 0.2, 0.5, 0.3$ , find $E(X)$ .

Excellence

A game costs $2 to play. You win $5 with probability 0.3 and nothing otherwise. Find the expected profit per game and state whether the game is worth playing.