Expected value

Expected value is the probability-weighted average of all possible outcomes of a random variable, representing what the variable averages to over many repetitions. Each outcome is multiplied by its probability and the products are summed.

Also known asexpectation

ByHoang TruongUpdated June 30, 2026

See it move

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The tree diagram shows E[X] = €5,250 at the root, branching via a summation operator into three probability-weighted outcomes: €12,000 at probability 0.25 contributes €3,000; €5,000 at probability 0.45 contributes €2,250; and €0 at probability 0.30 contributes €0. The three branches sum to the root value, making explicit that expected value is a probability-weighted average across all possible outcomes, not a prediction that any single outcome will occur.

Where it fits

TopicProbability & DistributionsCoreTopicDescriptive StatisticsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

E(X) = \sum_{i} x_i \cdot p_i

Variables

$x_{i}$: Possible value that the random variable can take
$p_{i}$: Probability that outcome xᵢ occurs; all pᵢ must sum to 1

For a discrete random variable; represents the long-run average outcome over many repetitions

Check yourself

PracticeCORE

A sales agent can close a deal worth €15,000 with probability 0.30, a deal worth €6,000 with probability 0.50, or no deal with probability 0.20. What is the expected revenue per sales call?

Select an answer to check your understanding.