Expected Value Calculator

Calculate the expected value E[X] of a discrete random variable from a list of outcomes and probabilities. Also shows variance and standard deviation.

Value x

Probability p(x)

E[X] = 1.7

Var(X) = 0.61

σ(X) = 0.781025

Σ p = 1

How to Calculate Expected Value

List each possible outcome xᵢ in the Value column.
Enter the probability p(xᵢ) for each outcome. Probabilities must sum to 1.
Click Add row to extend the table if you have more outcomes.
The calculator instantly shows E[X], variance, and standard deviation.

Anwendungsfälle

•Probability and statistics coursework.
•Gambling and game analysis — is a bet fair?
•Decision theory — pick the action with the best expected payoff.

Formel

E[X] = Σ xᵢ · p(xᵢ). Var(X) = E[X²] − (E[X])². σ(X) = √Var(X). Probabilities must sum to 1.

Häufig gestellte Fragen

What is expected value?

The expected value E[X] of a discrete random variable is the long-run average outcome, weighted by probabilities. It is computed as Σ xᵢ · p(xᵢ).

Do the probabilities need to sum to 1?

Yes. A valid probability distribution must satisfy Σ p(xᵢ) = 1. The calculator flags deviations beyond 0.001.

What do variance and standard deviation tell me?

Variance measures how spread out the values are around the mean. Standard deviation σ(X) is its square root, expressed in the same units as X.