Expected Value Proof Calculator | Discrete Outcome Solver

Enter Discrete Outcomes and Probabilities

Use the form below to compute expected value and verify the linearity proof for a transformed variable.

Outcome x1

Probability p1

Weighted Term x1p1

Outcome x2

Probability p2

Weighted Term x2p2

Outcome x3

Probability p3

Weighted Term x3p3

Outcome x4

Probability p4

Weighted Term x4p4

Outcome x5

Probability p5

Weighted Term x5p5

Outcome x6

Probability p6

Weighted Term x6p6

Scale Constant a

Used in the proof of E[aX + b] = aE[X] + b.

Shift Constant b

Changes the transformed variable without changing the original distribution.

Decimal Precision

Controls rounding for displayed metrics, proof steps, and exports.

Normalize probabilities automatically when total differs from 1

Example Data Table

This sample distribution is also available through the “Load Example” button above.

Outcome	Probability	Outcome × Probability
-2	0.08	-0.16
0	0.12	0.00
1	0.20	0.20
3	0.25	0.75
5	0.20	1.00
7	0.15	1.05
Total	1.00	2.84

Formula Used

Expected value: E[X] = Σ(xᵢpᵢ)

Expected square: E[X²] = Σ(xᵢ²pᵢ)

Variance: Var(X) = E[X²] - (E[X])²

Standard deviation: σ = √Var(X)

Linearity proof: E[aX + b] = Σ((axᵢ + b)pᵢ) = aΣ(xᵢpᵢ) + bΣpᵢ = aE[X] + b

Because the probabilities form a proper discrete distribution, Σpᵢ = 1. That makes the constant term simplify cleanly in the final proof step.

How to Use This Calculator

Enter each possible outcome in the outcome fields.
Enter the matching probability for every outcome.
Set constants a and b to test the transformed variable.
Choose a display precision for cleaner proof steps.
Enable normalization if your probabilities are raw weights or percentages.
Click Compute Expected Value Proof.
Review the summary metrics, proof lines, detail table, and Plotly graph.
Use the CSV or PDF buttons to export your result report.

FAQs

1. What does expected value represent?

Expected value is the weighted average of all possible outcomes. It shows the long-run average result you would expect after many repetitions of the same random process.

2. Why must probabilities sum to 1?

A valid discrete probability distribution must account for every possible outcome exactly once. Summing to 1 confirms the model covers the full event space.

3. Can I enter percentages instead of decimals?

Yes. Enter them as decimal equivalents, or enter raw weights and enable normalization. For example, 20% can be entered as 0.20.

4. What does the proof section verify?

It verifies the linearity rule for expectation. The calculator compares the direct computation of E[aX + b] with the theorem value aE[X] + b.

5. Can outcomes be negative or decimal values?

Yes. Outcomes may be negative, positive, zero, or fractional. The calculator accepts any numeric value that matches a valid probability.

6. Why are E[X²] and variance included?

They measure spread, not just center. Two distributions can share the same expected value while having very different levels of variability.

7. When should I enable normalization?

Enable it when your probability entries are proportional weights, rounded percentages, or values that almost sum to 1 but miss slightly.

8. What does the graph show?

The graph displays probabilities, weighted contributions xᵢpᵢ, and transformed contributions. It helps you see which outcomes drive the final expectation most strongly.