Poisson Distribution Calculator
This Poisson distribution calculator computes the probability of observing a given number of events k in a fixed interval when events occur with a known average rate (lambda). Use it to evaluate rare event probabilities, queueing problems, and many discrete stochastic processes.
Understanding the Poisson distribution is essential for statisticians, researchers, and professionals in various fields such as telecommunications, finance, and natural sciences, where the modeling of random events is crucial.
What is the Poisson Distribution?
The Poisson distribution models the probability of a number of events occurring in a fixed interval when events happen independently and at a constant average rate. It is parameterized by lambda (λ), the expected number of events in the interval.
P(X = k) = (e^-λ * λ^k) / k!The calculator supports three common probability queries:
- P(X = k): The probability of exactly k events (PMF).
- P(X ≤ k): The cumulative probability of at most k events (CDF).
- P(X ≥ k): The probability of at least k events (upper tail).
Example
Suppose on average 3 emails arrive per hour (λ = 3). What is the probability exactly 2 emails arrive in one hour?
Using the formula:
P(X = 2) = e^-3 * 3^2 / 2! = e^-3 * 9 / 2 = 0.2240418077So about 22.40% chance of exactly 2 emails arriving.
How to Use the Poisson Distribution Calculator
- Enter the average rate λ (lambda). Must be >= 0.
- Enter the number of events k you want the probability for (k must be an integer >= 0).
- Select the probability type: P(X = k), P(X ≤ k), or P(X ≥ k).
- The result updates automatically. Toggle charts to visualize the PMF across k values.
Use Cases of the Poisson Distribution
The Poisson distribution has many applications in real-world scenarios, including:
- Modeling the number of phone calls received at a call center per hour.
- Predicting the number of decay events per unit time from a radioactive source.
- Counting the number of customers arriving at a store within a given timeframe.
- Calculating the probability of errors occurring in a process or manufacturing step.
FAQ about Poisson Distribution
Can λ (lambda) be non-integer?
Yes. Lambda (λ) is the average rate and can be any non-negative real number.
What if k is large?
The calculator uses stable iterative methods to compute probabilities. For very large k, probabilities may underflow to values near zero; in such cases, consider using log-probabilities or approximations (e.g., normal approximation) if appropriate.
When is the Poisson model appropriate?
Use the Poisson model when events occur independently, with a constant average rate, and when two events cannot occur at exactly the same instant (or are rare). Typical applications include counts of arrivals, defects, or occurrences in fixed intervals of time or space.
What are the limitations of the Poisson distribution?
While the Poisson distribution is useful, it assumes that events are rare and independent. If events are not independent, or if the average rate changes significantly over time, this might not be the right model. For more complex scenarios, consider a more advanced statistical model.