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Expected Value (Discrete Random Variable)

Calculates the long-run average value of a discrete random variable, weighted by the probability of each possible outcome.

Written by the Vorzep Team

StatisticsProbabilityFundamental

Expected Value CalculatorDiscrete Random Variable

E(X) = Σ ( xᵢ · P(xᵢ) )
E(X) = expected value  ·  xᵢ = outcome  ·  P(xᵢ) = probability
⟹ Compute⟷E(X) from x & P(x)
Outcome (x)Probability P(x)Contribution
ΣP(x) =0E(X) =0
Presets:
Please fix the errors above.
Expected Value—
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E(X): —Var(X): —σ(X): —ΣP: —
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Expected Value Distribution—
E(X) location Distribution range
E(X) = Σ (xᵢ · P(xᵢ))  ·  ΣP(xᵢ) = 1  ·  Var(X) = E(X²) − [E(X)]²

Variables

SymbolQuantityUnit
E(X)Expected value
xᵢEach possible outcome
P(xᵢ)Probability of that outcome

What it means

The expected value (or expectation) of a discrete random variable is the weighted average of all possible values, weighted by their probabilities. It represents the long‑run average if the experiment is repeated many times. It is a measure of central tendency for random variables. In finance, it is used to calculate expected returns; in insurance, to set premiums; in decision theory, to evaluate choices. The expected value is a fundamental concept in probability and is used to define variance, covariance, and other moments. Understanding expectation is essential for risk analysis and for rational decision making under uncertainty.

Worked example

Expected Value (Discrete) – Two Examples

Real‑World
Scenario: A game: win $100 with prob 0.1, lose $10 with prob 0.9. Find expected value.
ParameterValue
Outcomes$100 (p=0.1), -$10 (p=0.9)
1E(X) = 100×0.1 + (−10)×0.9 = 10 − 9 = 1
Result $1 ✓ Expected win per play
Scenario: A fair die: outcomes 1-6 each with p=1/6. Find expected value.
ParameterValue
Outcomes1,2,3,4,5,6
1E(X) = (1+2+3+4+5+6)/6 = 21/6 = 3.5
Result 3.5 ✓ Average roll
Insight: Expected value is the long‑run average of a random variable. It is a weighted average of outcomes.

Common mistakes

  • Expected value: The long‑run average of a random variable.
  • Summation: Sum over all possible values of xᵢ · P(xᵢ).
  • Weights: The probabilities sum to 1.
  • Units: E(X) has the same units as X.
  • Linearity: E(aX+b) = aE(X)+b.

Applications

The expected value (mean) of a discrete random variable is the weighted sum of all possible values, weighted by their probabilities. It represents the long‑run average and is fundamental in decision theory, risk analysis, and engineering design. Engineers use expected value to evaluate design alternatives, to optimise resource allocation, and to assess financial risk. In quality control, it predicts average defect rates. By calculating expected value, professionals can quantify the central tendency of uncertain outcomes, making it a critical tool for decision‑making under uncertainty. It is also used in game theory, insurance, and project management. Understanding expected value enables objective comparison of alternatives based on their long‑run consequences.

  • Decision analysis – choosing among alternatives with uncertain outcomes
  • Risk assessment – expected losses or gains
  • Reliability – expected lifetime or number of failures
  • Inventory management – expected demand
  • Project management – expected project duration (PERT)