Poisson Distribution Calculator
The Poisson distribution models the number of times a rare event occurs in a fixed interval when events happen independently at a constant average rate λ. This calculator works accurately for k from 0 to 6.
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Formula
P(X = k) = (e^−λ × λᵏ) / k!
Raise e to the power of negative lambda (e^−λ), multiply by lambda raised to the power k (λᵏ), then divide by k factorial (k!). The factorial values used are: 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720. This calculator is designed for k between 0 and 6; for larger k values, use statistical software.
How to use the Poisson Distribution Calculator
- 1
Enter your average rate (λ)
- 2
Enter your number of events (k)
- 3
Read your results instantly
Results update in real time as you type.
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When to use the Poisson distribution
The Poisson distribution applies when you are counting the number of times a rare, independent event occurs in a fixed interval of time or space. Classic examples include: the number of phone calls arriving at a call center per minute, the number of cars passing a checkpoint per hour, the number of defects per meter of fabric, and the number of radioactive decay events per second.
For the distribution to be valid, events must be independent, the average rate (λ) must be constant, and two events cannot occur at exactly the same instant.
Calculator limitation: k from 0 to 6
This calculator uses hard-coded factorial values for k = 0 through 6. The formula engine does not support loops or factorial functions, so k values of 7 and above are not supported. For larger k values, use a scientific calculator, spreadsheet (POISSON.DIST in Excel), or statistical software like R (dpois).
For practical Poisson problems, k > 6 is relatively rare for small λ values. When λ = 3, the probability of k ≥ 7 is about 3.4% — manageable context for hand calculation or software.
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Poisson vs. binomial distribution
The Poisson distribution can be seen as the limit of the binomial distribution as n → ∞ and p → 0, with np = λ remaining constant. When n is large and p is small (rare events), Poisson is a good approximation for the binomial.
As a rule of thumb, use Poisson when n ≥ 20 and p ≤ 0.05 (or n ≥ 100 and p ≤ 0.10). Use the exact binomial when p is not small. The Poisson approximation is extremely convenient because it only requires λ = np, not the individual n and p values.
Tips & Insights
λ is both the mean and variance
A unique property of the Poisson distribution is that its mean equals its variance (both equal λ). If your count data has variance much larger than the mean, it is over-dispersed and Poisson may not be the right model.
Use k=0 for probability of zero events
P(X=0) = e^−λ gives the probability of no events occurring. For λ=3, that is e^−3 ≈ 0.0498, meaning about a 5% chance of zero events when the average is 3.
For cumulative probability, sum individual terms
To find P(X ≤ k), compute P(X=0) + P(X=1) + ... + P(X=k). For example, P(X ≤ 2) with λ=3 is about 0.0498 + 0.1494 + 0.2240 = 0.4232.
Worked Examples
Customer arrivals at a store
P(X=2) ≈ 0.1465 (14.65%). There is about a 15% chance that exactly 2 customers arrive in a given hour when the average rate is 4.
Equipment failures per month
P(X=0) ≈ 0.2231 (22.31%). There is about a 22% chance of zero failures in a month when the historical average is 1.5 failures.
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Frequently Asked Questions
What is the Poisson distribution?
The Poisson distribution gives the probability of a specific number of rare, independent events occurring in a fixed interval when the average rate λ is known.
What does λ (lambda) represent?
Lambda is the average rate of occurrence — the expected number of events in the interval. For example, if a call center receives an average of 5 calls per minute, λ = 5.
When is the Poisson distribution appropriate?
Use Poisson when events are rare, independent, and occur at a constant average rate in time or space. Examples include radioactive decay, insurance claims, equipment failures, and customer arrivals.
Why does this calculator only work for k = 0 to 6?
The formula engine requires factorial values to be hard-coded. Factorial grows rapidly (7! = 5040), and without a loop or recursive function, only small k values are directly supported. For k > 6, use statistical software.
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. The two distributions describe the same process from different perspectives.
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