What is a discrete probability distribution?

It's a list of all possible outcomes of a random variable along with the probability of each outcome occurring. The probabilities must be between 0 and 1 and must sum to exactly 1 across all outcomes.

Why does the calculator check that probabilities sum to 1?

A valid probability distribution must account for 100% of possible outcomes. If your probabilities sum to something other than 1, it signals a data-entry error or a missing outcome, so the calculator flags it instead of returning a misleading result.

What is the expected value (mean)?

The expected value, E(X), is the long-run average outcome you'd expect if you repeated the random process many times — calculated as the sum of each value multiplied by its probability: E(X) = Σ[x × P(x)].

What is variance in a probability distribution?

Variance measures how spread out the outcomes are from the expected value, calculated as Var(X) = Σ[P(x) × (x − E(X))²]. A larger variance means outcomes are more spread out; a smaller variance means they cluster near the mean.

How is standard deviation related to variance?

Standard deviation is the square root of the variance. It's expressed in the same units as the original values, making it easier to interpret than variance alone.

Can outcome values be negative?

Yes — outcomes can be any real number, including negative values, which is common for scenarios like profit/loss distributions where losses are represented as negative numbers.

How is this different from the Poisson or Binomial calculators?

Poisson and Binomial calculators compute probabilities from a fixed mathematical formula based on a couple of parameters. This calculator instead takes a distribution you define outcome-by-outcome and summarizes it — useful when your scenario doesn't match a standard named distribution.

Where are custom discrete distributions used in real life?

They're used to model lottery or game payouts, insurance claim amounts, survey response frequencies, and any scenario where outcomes and their likelihoods are known from data rather than a formula.

Probability Distribution Calculator — Analyze Any Discrete Distribution

Our Probability Distribution Calculator analyzes any discrete probability distribution you define — enter each possible outcome and its probability, and the calculator validates that the probabilities sum to 1, then computes the expected value (mean), variance, and standard deviation using the standard weighted-sum formulas. Unlike the site's Poisson, Binomial, and Negative Binomial calculators, which model specific named distributions, this tool works with any custom distribution you construct from scratch.

Quick Answer

Enter each outcome and its probability as 'value,probability' pairs, one per line. This calculator validates that your probabilities sum to 1, then computes the expected value (mean), variance, and standard deviation of your custom discrete distribution.

Distribution (value,probability per line)

Enter one outcome per line as 'value,probability' — probabilities must sum to 1.

How to Use the Probability Distribution Calculator — Mean, Variance & Std Dev

1
Enter each possible outcome and its probability as one 'value,probability' pair per line.
2
Make sure your probabilities add up to 1 (the calculator will flag it if they don't).
3
Click 'Calculate' to get the expected value (mean), variance, and standard deviation.
4
Review the step-by-step formulas showing exactly how each statistic was computed.

Why Use Probability Distribution Calculator — Mean, Variance & Std Dev?

Not every probability scenario fits a named distribution like Poisson or Binomial — sometimes you just have a custom table of outcomes and their probabilities, like a game's payout structure or a survey's response distribution, and need the expected value and spread. Computing E(X) = Σ[x × P(x)] and Var(X) = Σ[P(x) × (x − E(X))²] by hand across many rows is tedious and easy to get wrong, especially when the probabilities don't quite sum to 1 due to a data-entry mistake. This calculator validates your inputs and computes every statistic instantly.

Frequently Asked Questions

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