Poisson Process Model Calculator

Analyze rate, time, and event thresholds easily. Compare exact, cumulative, tail, and interval event probabilities. Export polished results and visualize distributions with interactive charts.

Calculator Inputs

Use manual rate mode when λ is known. Use estimate mode when you have observed interval counts and want λ inferred automatically.

Choose how the event rate should be supplied.
Events per unit time, such as calls per hour.
The forecast window for event-count probabilities.
Used for exact, cumulative, and tail calculations.
Minimum event count for the interval range.
Maximum event count for the interval range.
Controls the x-axis upper limit of the chart.
Choose the number of decimal places to show.
Required only when estimating λ from equal intervals.
Enter non-negative whole numbers separated by commas, spaces, or semicolons.
Reset

Formula Used

1) Interval mean

μ = λt

λ is the event rate per unit time. t is the chosen interval length.

2) Exact event probability

P(N(t) = k) = e-μ μk / k!

This gives the probability of exactly k events in the interval.

3) Cumulative and interval probabilities

P(N(t) ≤ k) = Σ P(N(t) = i), for i = 0 to k

P(a ≤ N(t) ≤ b) = P(N(t) ≤ b) − P(N(t) ≤ a−1)

4) Mean and variance

E[N(t)] = μ

Var(N(t)) = μ

SD(N(t)) = √μ

5) Waiting times

E[T₁] = 1 / λ

E[Tk] = k / λ

The first waiting time is exponential. The k-th waiting time follows an Erlang model.

How to Use This Calculator

  1. Choose Manual λ when you already know the event rate, or select Estimate λ from data if you have observed counts.
  2. Enter the time interval t and the event thresholds k, a, and b.
  3. Set the graph range and preferred decimal precision, then submit the form.
  4. Review the result summary, detailed table, downloadable exports, and distribution graph above the form.

Example Data Table

This sample uses a mean of μ = 2.4 events per one-hour interval. It helps illustrate how observed counts compare with model probabilities.

Interval Observed Count Expected Mean μ Exact Probability P(N = observed)
Hour 1 1 2.4000