Calculator
Example Data Table
| Outcome | Probability | x × p(x) |
|---|---|---|
| 0 | 0.10 | 0.00 |
| 1 | 0.20 | 0.20 |
| 2 | 0.30 | 0.60 |
| 3 | 0.25 | 0.75 |
| 4 | 0.15 | 0.60 |
| Total | 1.00 | 2.15 |
Formula Used
For a discrete random variable, the expected value is the sum of each outcome multiplied by its probability.
E(X) = Σ[x × p(x)]
The second moment is also helpful.
E(X²) = Σ[x² × p(x)]
Variance measures spread around the expected value.
Var(X) = E(X²) − [E(X)]²
Standard deviation is the square root of variance.
σ = √Var(X)
If you enable normalization, the calculator first converts each entered probability by dividing it by the total probability.
p′(x) = p(x) / Σp(x)
How to Use This Calculator
- Enter one outcome and one probability on each row.
- Use decimals, percentages, or fractions for probabilities.
- Choose decimal places for the displayed output.
- Enable normalization if your probabilities do not sum to one.
- Keep outcome sorting on if you want cumulative probability in order.
- Press the calculate button to show the summary above the form.
- Review the row contributions, variance, and spread measures.
- Download the result as CSV or PDF when needed.
About Expected Value in Probability Distribution
What Expected Value Means
Expected value is the long run average of a discrete random variable. It tells you the weighted center of all possible outcomes. Each outcome is multiplied by its probability. Then the products are added. This calculator speeds up that process and reduces manual mistakes.
Why Distribution Quality Matters
A probability distribution must be valid before interpretation. The probabilities should be nonnegative and should sum to one. When entries do not sum exactly to one, results may mislead. That is why this page checks the total probability first. It can also normalize the values when you allow it.
Where This Calculator Helps
This calculator is useful in statistics, finance, insurance, quality control, games, and decision analysis. You can study payoffs, expected scores, expected profit, or expected loss. You can also compare two scenarios with different risks. The extra outputs make the tool more practical. It shows variance, standard deviation, cumulative probability, and a row by row contribution table.
Core Measures Behind the Output
The main formula is simple. For a discrete variable X, expected value equals the sum of x multiplied by p of x. Variance equals the sum of squared outcomes times probability, minus the square of the expected value. Standard deviation is the square root of variance. These measures work together. Expected value shows the center. Variance and standard deviation show spread.
Using the Tool Well
To use the calculator, enter each outcome and its probability on separate rows. You may type decimals, percentages, or simple fractions. Choose the number of decimal places you want in the output. You can also sort the outcomes or normalize the probabilities automatically. After submitting, the result appears above the form. You can then review the summary, inspect the distribution table, and export the data.
The example table helps you test the page quickly. It is also useful for classroom demonstrations and assignment checks. Because the layout is clean and responsive, the form stays readable on desktop, tablet, and mobile screens.
Reading the Result Carefully
Remember that expected value does not guarantee a likely single result. Real observations can differ. A negative expected value suggests an average loss over many trials. A positive expected value suggests an average gain. Still, large variance means outcomes may swing widely. Always inspect both center and spread before making a decision. This keeps interpretation balanced and statistically sound.
FAQs
1. What does expected value mean?
Expected value is the weighted average of all possible outcomes. Each outcome is multiplied by its probability. The products are then added.
2. Can I use this for any probability model?
Use it only for discrete distributions. Each row should represent one possible value and one probability. For continuous models, use an integral based method instead.
3. Can I enter percentages or fractions?
Yes. Percentages such as 25% and fractions such as 1/4 can be entered. The calculator converts them into decimal probabilities before calculation.
4. What makes a distribution valid?
A valid probability distribution has probabilities that are not negative and add up to one. If the total is off, enable normalization or correct the entries.
5. Why are variance and standard deviation shown?
Variance measures spread around the expected value. Standard deviation is the square root of variance, so it is easier to read in original outcome units.
6. May outcomes be negative?
Yes. Outcomes may be negative, zero, or positive. This is common in profit and loss problems, game payoffs, and scoring models.
7. What do the CSV and PDF buttons do?
CSV export saves the summary and row data in a spreadsheet friendly file. PDF export creates a simple report that is easy to print or share.
8. Where will the answer appear?
The result appears above the form after submission. This keeps the summary visible while you review or edit the distribution entries below.