What is expected value for a discrete random variable?

The expected value E(X) = Σ xᵢ × pᵢ is the probability-weighted average of all possible outcomes. It represents the long-run average if the experiment were repeated many times.

Why must probabilities sum to 1?

A valid probability distribution requires that the total probability of all outcomes equals exactly 1, meaning all possible outcomes are accounted for. The calculator warns you if the sum deviates from 1.

Can I enter probabilities as fractions like 1/6?

Yes. The calculator parses fractions (e.g. 1/6) and decimals (e.g. 0.1667) automatically. Both formats are accepted in the probability fields.

What is the difference between variance and standard deviation?

Variance Var(X) = Σ pᵢ(xᵢ − E(X))² measures spread in squared units. Standard deviation σ = √Var(X) converts it back to the original units, making it easier to interpret.

Can outcome values be negative?

Yes. Outcome values can be any real number — negative, zero, or positive. Only probabilities are constrained to be between 0 and 1.

Expected Value Calculator (Discrete)

Name: Expected Value Calculator (Discrete)
Availability: InStock
Author: Nham Vu

Enter outcome-probability pairs to compute E(X), Var(X), and standard deviation for any discrete random variable.

Probability Distribution

Value (xᵢ)

Probability (pᵢ)

Sum of probabilities: 0.0000

Quick Presets

Results

E(X)

—

Expected Value

Var(X)

—

Variance

σ(X)

—

Std Deviation

Number of outcomes: —
Probability sum: —

Term-by-Term Breakdown

Enter values and click Calculate to see breakdown.

xᵢ	pᵢ	xᵢ×pᵢ	(xᵢ−E)²	pᵢ(xᵢ−E)²

Summary

Enter outcome-probability pairs to compute E(X), Var(X), and standard deviation for any discrete random variable.

How it works

Enter each outcome value (xᵢ) and its probability (pᵢ) in the table rows.
Click "Add Row" to add more outcome-probability pairs.
The probability sum indicator turns green when all probabilities sum to exactly 1.
Click "Calculate" to compute E(X), Var(X), and standard deviation.
Review the breakdown table to see each term's contribution to the expected value and variance.

Use cases

Computing the expected payoff of a lottery or gambling game.
Finding the average outcome of a custom probability experiment.
Verifying textbook discrete distribution problems.
Calculating expected profit or loss from a decision tree.
Teaching statistics: building intuition for weighted averages.
Quality control: expected number of defects per batch.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu