Random Variables and Expectation

≈ 20 min

A random variable $X$ assigns a number to each outcome of an experiment — for instance, the number shown on a die, or your winnings in a game. Its probability distribution lists each value with its probability, and those probabilities sum to 1.

The expected value $E[X]$ is the long-run average — each value weighted by its probability:

E[X] = \sum x \cdot P(x).

Worked example. For a fair die, $E[X] = \dfrac{1+2+3+4+5+6}{6} = \dfrac{21}{6} = 3.5$ . You will never roll a 3.5, but over thousands of rolls the average settles there.

The expected value $E[X]$ of a random variable is best described as:

the most likely single outcome

the long-run average value over many repetitions

the largest possible value

a value $X$ can always actually equal

A fair six-sided die is rolled. What is the expected value of the number shown?

A random variable $X$ takes value 0 with probability 0.5, value 1 with probability 0.3, and value 2 with probability 0.2. Find $E[X]$ .

A school raffle ticket costs R5. There is a 0.05 probability of winning the R50 prize and a 0.95 probability of winning nothing. What is the expected profit (in rand) from buying one ticket? (A loss is negative.)