Enter Distribution Inputs
Example Data Table
Use this sample distribution to test the calculator before entering your own values.
| Outcome x | Probability P(X=x) | Interpretation |
|---|---|---|
| 0 | 0.10 | No event occurs. |
| 1 | 0.20 | One event occurs. |
| 2 | 0.30 | Two events occur. |
| 3 | 0.25 | Three events occur. |
| 4 | 0.15 | Four events occur. |
Formula Used
Probability mass function: P(X = xi) = pi, where all probabilities are nonnegative and sum to 1.
Normalization for weights: pi = wi / Σwi.
Expected value: E[X] = Σxipi.
Second moment: E[X²] = Σxi2pi.
Variance: Var(X) = E[X²] − (E[X])².
Standard deviation: σ = √Var(X).
Cumulative distribution: F(x) = P(X ≤ x) = Σpi for all outcomes not exceeding x.
Range probability: P(a ≤ X ≤ b) = Σpi for outcomes between a and b, inclusive.
Set probability: P(X ∈ S) = Σpi for every outcome contained in the selected set S.
How to Use This Calculator
1. Enter each discrete outcome in the outcome box. Keep the order consistent with your probabilities or weights.
2. Choose exact probabilities if your numbers already sum to 1. Choose weights when your entries are relative frequencies or scores.
3. Add a target value to evaluate equality, less-than, and greater-than probabilities.
4. Enter lower and upper bounds to calculate an inclusive interval probability.
5. Add specific set values when you want the probability of selected outcomes only.
6. Press the calculate button. The result block appears under the header and above the form.
7. Export the generated distribution table and metrics using the CSV or PDF buttons.
FAQs
1. What does this calculator measure?
It evaluates a discrete probability distribution. You can calculate exact outcome probabilities, cumulative probabilities, expected value, variance, standard deviation, entropy, and interval or set-based probabilities from custom inputs.
2. What is a discrete probability distribution?
A discrete distribution lists separate countable outcomes and assigns a probability to each one. Examples include dice totals, defect counts, arrivals, purchases, or any variable that takes individual numeric values.
3. When should I use weights instead of probabilities?
Use weights when your values do not yet sum to 1. The calculator converts each weight into a probability by dividing it by the total weight, which preserves relative importance.
4. Why must probabilities sum to 1?
The total probability across all possible outcomes must equal certainty. If the sum differs from 1, the distribution is incomplete or inconsistent, so exact probability mode rejects it.
5. What does the expected value mean?
Expected value is the long-run weighted average outcome. It does not need to be one of the listed outcomes, but it summarizes the center of the distribution effectively.
6. What is the difference between PMF and CDF?
The PMF gives the probability of each exact outcome. The CDF adds probabilities cumulatively and shows the chance that the random variable is less than or equal to a chosen value.
7. Can I use decimals and negative outcome values?
Yes. Outcome values may be decimals or negative numbers. Only probabilities or weights must stay nonnegative. Negative outcomes are common in finance, scoring systems, and net-change models.
8. What do the CSV and PDF exports include?
The exports capture the generated result metrics and the distribution table after calculation. This helps with documentation, classroom review, audit trails, reporting, or sharing results with others.