Math

Poisson Distribution Calculator

Calculate Poisson probabilities for any rate and event count

Poisson Distribution Calculator

The Poisson distribution models the number of events that occur in a fixed interval of time, distance, or space — when events happen independently and at a constant average rate. Classic examples include website visits per minute, defects per meter of wire, emails per hour, and bacteria colonies per petri dish.

This calculator computes exact probabilities and cumulative probabilities for the Poisson distribution given a rate parameter λ (lambda) and an observed count k.

How to Use This Calculator

  1. Enter λ (lambda) — the average number of events per interval. For example, if calls arrive at 4 per hour, λ = 4.
  2. Enter k — the number of events you want to calculate a probability for (whole number ≥ 0).
  3. Select mode — "exactly k", "at most k" (CDF), or "at least k" (survival function).
  4. The calculator returns the probability P(X = k), P(X ≤ k), or P(X ≥ k) along with the mean and standard deviation.

Poisson Probability Formula

The probability mass function is:
P(X = k) = e-λ × λk / k!
where e ≈ 2.71828 is Euler's number.

The mean equals λ, and uniquely for the Poisson distribution, the variance also equals λ. The standard deviation is √λ.

Real-World Examples

Call center: A support line receives 6 calls per hour on average. What is the probability of exactly 4 calls in any given hour? λ = 6, k = 4 → P ≈ 0.1339 (13.4%).

Traffic accidents: A stretch of road averages 1.2 accidents per week. What is the probability of at least 2 accidents in a week? λ = 1.2, k = 2, mode = "at least" → P ≈ 0.337.

Radioactive decay: A sample emits 3 particles per second. What is the probability of zero emissions in a second? λ = 3, k = 0 → P = e-3 ≈ 0.0498.

Poisson vs. Binomial Distribution

The Poisson is derived as the limiting case of the binomial as n → ∞ and p → 0, keeping λ = np constant. When n ≥ 20 and p ≤ 0.05, use Poisson with λ = np for a simpler calculation. When the number of trials is known and fixed, use the binomial instead.

Frequently Asked Questions

What does lambda (λ) mean in the Poisson distribution?
Lambda is the average rate (expected count) of events in the interval of interest. If a server receives 200 requests per minute, λ = 200. Lambda must be positive.
Can k be 0 in the Poisson distribution?
Yes. P(X = 0) = e^(-λ), the probability of no events occurring. For λ = 3, that is e^(-3) ≈ 4.98%. This is often the probability you want to minimize (e.g., zero sales).
When should I use Poisson instead of binomial?
Use Poisson when: events occur in a continuous interval (time, length, area), the number of possible events has no fixed upper bound, and events are rare relative to the opportunity. Use binomial when n (total trials) is fixed and small.
Why does the Poisson distribution have equal mean and variance?
This is a mathematical property of the Poisson process. It means the standard deviation equals √λ. In practice, if your observed variance is much larger than the mean, your data may follow a negative binomial distribution (overdispersion) instead.
What is overdispersion, and should I worry about it?
Overdispersion means the variance exceeds the mean — a sign that events are not truly independent or the rate varies over time. If your count data shows this pattern, a negative binomial or zero-inflated model may be more appropriate than Poisson.
What does "at most k" mean in the Poisson calculator?
"At most k" computes P(X ≤ k) = Σ P(X = i) for i = 0 to k. This is the cumulative distribution function (CDF) and answers questions like "what is the chance of 5 or fewer events?"
Can I use the Poisson distribution for large k?
Yes. When λ is large (λ > 30), the Poisson distribution is well-approximated by a normal distribution with mean λ and standard deviation √λ. This calculator uses log-space computation for accurate results even for large λ and k.
How is the Poisson distribution related to the exponential distribution?
If events follow a Poisson process with rate λ, the time between consecutive events follows an exponential distribution with mean 1/λ. So if calls arrive at 4 per hour (λ = 4), the average wait between calls is 1/4 hour = 15 minutes.

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