Random Variable Expected Value Calculator

Calculator Input

Use the form below to estimate the expected value of a discrete random variable. The page remains single column, while the input grid changes across screen sizes.

Probability Input Mode Decimal values Percent values

Decimal Places

Sort Output Keep input order Sort by outcome ascending Sort by outcome descending

Normalize probabilities automatically when totals do not equal 1

Label

Outcome (x)

Probability

Remove Row

Label

Outcome (x)

Probability

Remove Row

Label

Outcome (x)

Probability

Remove Row

Label

Outcome (x)

Probability

Remove Row

Add Row Load Example Calculate Expected Value Reset Form

Example Data Table

This sample uses a valid discrete distribution because the probabilities sum to 1. The expected value equals the sum of all outcome-probability products.

Formula Used

For a discrete random variable, the expected value is the weighted average of all possible outcomes.

Expected value: E(X) = Σ[x × p(x)]

Second moment: E(X²) = Σ[x² × p(x)]

Variance: Var(X) = E(X²) - (E(X))²

Standard deviation: σ = √Var(X)

If the entered probabilities do not total 1, you can normalize them. Normalization divides each probability by the total sum before calculation.

How to Use This Calculator

1. Enter one row for each outcome of the random variable.
2. Add an optional label if you want named events.
3. Enter probabilities as decimals or percentages.
4. Choose whether to normalize totals automatically.
5. Set decimal places and preferred output order.
6. Press the calculate button.
7. Review the expected value, variance, table, and graph above the form.
8. Export the result table as CSV or PDF when needed.

Frequently Asked Questions

1. What does expected value mean?

Expected value is the long-run average outcome of a discrete random variable. It weights each possible value by its probability, then adds those weighted outcomes together.

2. Do probabilities need to sum to 1?

Yes. A valid discrete probability distribution totals 1. This calculator can also normalize entries automatically when the checkbox is enabled.

3. Can I enter percentages instead of decimals?

Yes. Choose percent mode, then enter values like 25 for 25%. The calculator converts them into decimal probabilities before computing the expected value.

4. Why is my expected value not one of the outcomes?

Expected value is a weighted average. Averages often fall between listed outcomes, so the result does not need to match any single possible value.

5. What is the difference between variance and standard deviation?

Variance measures weighted spread in squared units. Standard deviation is the square root of variance, so it returns spread in the original outcome units.

6. Can this calculator handle negative outcomes?

Yes. Outcomes may be positive, zero, or negative. The only restriction is that probabilities cannot be negative and must form a valid total.

7. When should I normalize probabilities?

Normalize when your entries are meant to represent relative likelihoods but do not total 1 due to rounding or partial raw inputs.

8. What does the chart show?

The chart plots the discrete probability distribution. It helps you see which outcomes carry more probability mass before interpreting the expected value.