Calculators
Poisson Distribution Calculator
Calculate Poisson probabilities, mean, variance, and standard deviation for a given average rate locally in your browser.
Enter the average rate (λ) and number of events (k). Calculations run locally in your browser; your values are not uploaded or stored.
Find exact and cumulative probabilities for Poisson processes.
A Poisson process models events occurring randomly at a constant average rate λ over a fixed interval. Enter the average rate λ and a target number of events k. The calculator returns the mean λ, variance λ, standard deviation √λ, and exact probability P(X=k), plus cumulative probabilities P(X≤k), P(X<k), P(X>k), and P(X≥k). All calculations use stable logarithmic arithmetic and run locally in your browser; your values are not uploaded or stored.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is a Poisson distribution?
A Poisson distribution models the number of events occurring in a fixed interval of time or space, given a constant average rate λ and independence between events. It is defined by a single parameter λ (lambda), the average rate of occurrence. Common examples include the number of emails received per hour, cars passing a checkpoint per minute, or defects in a manufactured batch.
What formulas does this calculator use?
The calculator uses: Mean = λ, Variance = λ, Standard deviation = √λ. The probability mass function is P(X = k) = e^(-λ) × λ^k / k!. Cumulative probabilities sum the PMF: P(X ≤ k) = Σ_{i=0}^{k} P(X = i). All calculations use logarithmic arithmetic to handle large λ without overflow.
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the exact probability of observing exactly k events. P(X ≤ k) is the cumulative probability of observing k or fewer events, which is the sum of P(X = i) for all i from 0 to k.
When should I use a Poisson distribution instead of a binomial distribution?
Use a Poisson distribution when events occur continuously at a known average rate but the number of trials is not fixed (or is very large). Use a binomial distribution when you have a fixed number of independent trials with a constant success probability. The Poisson distribution can approximate the binomial when n is large and p is small, with λ = n × p.
Do my values stay private?
Yes. All calculations run locally in your browser. CodeASystem does not upload or store the values you enter.