Poisson distribution calculator
Enter your values. The answer and the full worked solution update as you type.
Result
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.01832 | 0.01832 |
| 1 | 0.07326 | 0.09158 |
| 2 | 0.1465 | 0.2381 |
| 3 | 0.1954 | 0.4335 |
| 4 | 0.1954 | 0.6288 |
| 5 | 0.1563 | 0.7851 |
| 6 | 0.1042 | 0.8893 |
| 7 | 0.05954 | 0.9489 |
| 8 | 0.02977 | 0.9786 |
| 9 | 0.01323 | 0.9919 |
| 10 | 0.005292 | 0.9972 |
| 11 | 0.001925 | 0.9991 |
| 12 | 0.0006415 | 0.9997 |
Find Poisson probabilities for the number of events in a fixed interval, such as calls per hour, defects per metre or arrivals per minute. Enter the average rate λ and a count x to get the probability of exactly x events, at most x, at least x, fewer than x or more than x. The steps show the formula e^(−λ)·λˣ/x! for each term and use the complement where that is shorter.
Step-by-step solution
Poisson probability formula
Use the complement (fewer terms)
Mean and variance
Key formulas
- Complement
- P(not A) = 1 − P(A)
How to calculate Poisson probabilities
- Find λ, the average number of events in the interval the question is about. If the rate is 3 per hour and the question is about 2 hours, use λ = 6.
- For exactly x events use P(X = x) = e^(−λ)·λˣ/x!.
- For “at most x”, add the probabilities from 0 to x.
- For “at least x”, use the complement P(X ≥ x) = 1 − P(X ≤ x − 1), because the sum to infinity cannot be written out.
- Check your answer with the mean and the variance: both are equal to λ.
Questions and answers
When does a Poisson distribution apply?
When events happen one at a time, independently of each other and at a constant average rate, and you count how many fall in a fixed interval of time, length, area or volume.
What are the mean and variance of a Poisson distribution?
Both are equal to λ, so the standard deviation is √λ. If the variance of your data is much larger than the mean, the events are probably clustered and the Poisson model fits poorly.
Can λ be a decimal?
Yes. λ is an average, so values such as 2.5 events per hour are fine. The number of events x must be a whole number.
How is the Poisson distribution related to the binomial distribution?
For a binomial distribution with many trials and a small probability of success, the Poisson distribution with λ = np gives almost the same probabilities. A common rule of thumb is n ≥ 20 and p ≤ 0.05.
Why is P(X ≥ 1) equal to 1 − e^(−λ)?
The only way to have fewer than one event is to have none, and P(X = 0) = e^(−λ). So P(X ≥ 1) = 1 − e^(−λ), which is 0.8647 for λ = 2.
Practise it yourself
Practise Probability
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