Binomial & Poisson Calculator
PMF, CDF, mean and variance for binomial and Poisson distributions.
Overview
The Binomial and Poisson Calculator computes the probability mass function (PMF) and cumulative distribution function (CDF) for the two most-used discrete probability distributions, plus their mean and variance. Enter the parameters and a value of k and read off P(X = k), P(X <= k) and the distribution summary.
It is built for statistics students checking textbook answers, QA engineers modelling defect counts and product analysts estimating the chance of k events in a fixed window — like emails, signups or server errors.
How it works
For a binomial distribution with n trials and success probability p the PMF is P(X = k) = C(n, k) * p^k * (1 - p)^(n - k) for k = 0..n. Mean is n * p, variance is n * p * (1 - p). The CDF is the running sum.
The Poisson distribution describes counts in a fixed interval with mean λ. Its PMF is P(X = k) = e^-λ * λ^k / k!, and both its mean and variance equal λ. When n is large and p is small with λ = n * p, the binomial converges to Poisson.
Examples
Binomial(n=10, p=0.3), k=3 → PMF ≈ 0.2668, CDF ≈ 0.6496
Binomial(n=20, p=0.5), k=10 → PMF ≈ 0.1762, CDF ≈ 0.5881
Poisson(λ=4), k=2 → PMF ≈ 0.1465, CDF ≈ 0.2381
Poisson(λ=100), k=110 → PMF ≈ 0.0237, CDF ≈ 0.8540
FAQ
When should I use Poisson instead of binomial?
Use Poisson when events occur in continuous time or space with a known rate, and the number of trials is effectively unlimited. Use binomial when there is a fixed number of trials each with a fixed success probability.
Why is the Poisson variance always equal to the mean?
That's a defining property of the distribution. If your data has variance much larger than its mean, Poisson is the wrong model — consider negative binomial.
Can I compute P(X > k)?
Yes — it is 1 - CDF(k). The tool reports CDF so the complement is one subtraction away.
What if n*p is large?
Both distributions are well-approximated by a normal with the same mean and variance — a useful shortcut for back-of-envelope work.
Does it handle very small p?
Yes, but check that n * p stays in a sensible range. For very rare events Poisson is more numerically friendly.