Poisson Distribution Calculator
Calculate Poisson probabilities for rare event modeling.
Compute Poisson probabilities: P(X = k) and P(X ≤ k)
What Is the Poisson Distribution Calculator?
The Poisson Distribution Calculator computes the probability of exactly k events occurring in a fixed interval, given that events happen independently at a known average rate λ (lambda). It also computes the cumulative probability P(X ≤ k) and shows the distribution statistics.
Formula
How to Use
Enter λ (the average number of events per interval, must be > 0) and k (the specific number of events you want the probability for, a non-negative integer). Click Calculate to get P(X = k), P(X ≤ k), and the distribution stats.
Example Calculation
A call center receives on average 8 calls per hour (λ=8). What is the probability of exactly 5 calls in an hour? P(X=5) = e⁻⁸ × 8⁵ / 5! = 0.000335 × 32768 / 120 ≈ 9.16% P(X≤5) ≈ 19.12% Mean = 8, Std Dev = √8 ≈ 2.83
Understanding Poisson Distribution
The Poisson distribution, introduced by Siméon Denis Poisson in 1837, is named for its surprising property: the variance equals the mean. This makes it particularly useful for modeling count data where events are rare but the number of opportunities is large.
Poisson processes appear throughout science and engineering: in quantum mechanics (photon arrival), neuroscience (action potential firing), telecommunications (packet arrival in networks), and epidemiology (disease incidence rates).
The Poisson distribution also serves as an approximation when the binomial is computationally expensive: if n ≥ 20 and p ≤ 0.05, the Poisson with λ = np gives a very accurate approximation to B(n, p).
Frequently Asked Questions
When should I use the Poisson distribution?
Use Poisson when counting events that occur independently in a fixed time, area, or volume at a known average rate. It applies when: events are rare relative to possible opportunities, events are independent, and the rate is constant.
What is the relationship between Poisson and Binomial?
The Poisson distribution is the limiting case of the Binomial as n → ∞ and p → 0, with np = λ remaining constant. When n is large and p is small, Poisson approximates Binomial well.
What are real-world examples of Poisson processes?
Number of car accidents per day on a highway, number of radioactive decays per second, number of typos per page, number of customer arrivals per minute, number of mutations per DNA strand.
Can λ be a non-integer?
Yes. λ is an average rate and can be any positive real number (e.g., 2.5 calls per minute). Only k (the number of occurrences) must be a non-negative integer.
Why is Mean = Variance for Poisson?
This is a unique and defining property of the Poisson distribution. The mean and variance are both equal to λ. This equality can be used to test whether data follows a Poisson distribution.