Probability Distribution Calculator (Binomial, Poisson & more)

About the Probability Distribution Calculator (Binomial, Poisson & more)

Pick a distribution — binomial, Poisson, geometric, negative binomial, hypergeometric, discrete uniform, continuous uniform, exponential, normal, Student’s t, chi-square or F — type its parameters and get P(X = x), P(X ≤ x), P(X < x), P(X ≥ x), P(X > x), the probability between two values, or a quantile (the x with P(X ≤ x) = p, such as a critical value), with the mean, variance, standard deviation, median and mode and a chart with the probability shaded. Discrete distributions also get a table of P(X = x), P(X ≤ x) and P(X > x).

You can type your own discrete distribution as a table of values and probabilities (or counts) and get its expected value and variance — exactly, as fractions, when you type fractions. For count distributions the page also shows the normal approximation with a continuity correction next to the exact answer, with the usual rule of thumb for whether it can be trusted.

Common uses

• Checking statistics homework on binomial, Poisson, geometric and hypergeometric probabilities.
• Finding t, chi-square and F critical values for hypothesis tests and confidence intervals without a printed table.
• Computing the expected value and variance of a game, an insurance payout or any random variable given as a table.
• Seeing when the normal approximation to the binomial or Poisson distribution is good enough.

Which distribution fits

• Binomial(n, p): the number of successes in n independent trials that each succeed with probability p.
• Poisson(λ): the number of events in a fixed interval when they happen independently at an average rate λ.
• Geometric(p): the number of trials until the first success — or, in the other definition, the number of failures before it.
• Negative binomial(r, p): the number of failures before the r-th success, or the number of trials until it.
• Hypergeometric(N, K, n): the number of successes in n draws without replacement from N items of which K are successes.
• Uniform: every value (discrete) or every interval of the same length (continuous) equally likely. Exponential(λ): the waiting time between events of a Poisson process.
• Normal(μ, σ), t(ν), χ²(k), F(d₁, d₂): the bell curve and the three distributions behind most tests on means and variances.

How the probabilities are computed

Probabilities of single values come from the formula of each distribution, worked in logarithms so large n do not overflow. Cumulative probabilities use the regularized incomplete beta and gamma functions (NIST DLMF chapter 8): for a binomial P(X ≤ k) = I₁₋ₚ(n − k, k + 1), for a Poisson P(X ≤ k) = Q(k + 1, λ), for a chi-square P(X ≤ x) = P(k/2, x/2), for t and F the incomplete beta function; a hypergeometric sum is added from its end nearer the mode, each term from the one before it, until the terms are too small to matter. An upper tail is always computed from that tail itself rather than as 1 minus a number close to 1, so very small probabilities keep their digits. Quantiles of discrete distributions are the smallest k with P(X ≤ k) ≥ p; for continuous ones the cumulative distribution function is inverted numerically.

The formulas and parameter conventions follow the NIST/SEMATECH Gallery of Distributions.

The normal approximation and the continuity correction

A count X with mean μ and standard deviation σ can be approximated by a normal distribution when σ is not too small; because X takes whole-number values, each value k is spread over the interval from k − 0.5 to k + 0.5. So P(X ≤ k) ≈ Φ((k + 0.5 − μ)/σ), P(X ≥ k) ≈ 1 − Φ((k − 0.5 − μ)/σ) and P(X = k) ≈ Φ((k + 0.5 − μ)/σ) − Φ((k − 0.5 − μ)/σ). For a binomial the usual rule is np ≥ 5 and n(1 − p) ≥ 5 (some books use 10); for a Poisson, λ ≥ 10.

Limitations

• Parameters are real numbers typed as decimals or fractions; the binomial allows up to 10,000,000 trials and the hypergeometric a population of up to 10,000,000.
• Probabilities are shown to 6 significant digits and other values to 10; with parameters in the millions, such as a binomial with 10,000,000 trials, about 8 digits are reliable. Probabilities smaller than about 10⁻³⁰⁰ show as 0.
• When the possible values are very many or unbounded, the table and the chart cover those within 4.5 standard deviations of the mean (one value in every few when that is more than 200 values) and say how much of the probability that range holds.
• A custom table can have up to 500 values; its probabilities must add up to 1 unless you mark them as frequencies.

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