A Poisson distribution calculator is a powerful tool used to compute probabilities for events that occur randomly over a fixed interval of time or space, given a known average rate.
Calculating the probability of receiving 3 customer complaints in a day, given an average of 2 complaints per day.
Poisson Distribution Calculator
| λ (Average) | k (Events) | P(X = k) |
|---|---|---|
| 0.5 | 0 | 0.6065 |
| 0.5 | 1 | 0.3033 |
| 0.5 | 2 | 0.0758 |
| 1 | 0 | 0.3679 |
| 1 | 1 | 0.3679 |
| 1 | 2 | 0.1839 |
| 1.5 | 0 | 0.2231 |
| 1.5 | 1 | 0.3347 |
| 1.5 | 2 | 0.2510 |
| 2 | 0 | 0.1353 |
| 2 | 1 | 0.2707 |
| 2 | 2 | 0.2707 |
| 3 | 0 | 0.0498 |
| 3 | 1 | 0.1494 |
| 3 | 2 | 0.2240 |
| 3 | 3 | 0.2240 |
| 4 | 0 | 0.0183 |
| 4 | 1 | 0.0733 |
| 4 | 2 | 0.1465 |
| 4 | 3 | 0.1954 |
| 4 | 4 | 0.1954 |
| 5 | 0 | 0.0067 |
| 5 | 1 | 0.0337 |
| 5 | 2 | 0.0842 |
| 5 | 3 | 0.1404 |
| 5 | 4 | 0.1755 |
| 5 | 5 | 0.1755 |
| 6 | 0 | 0.0025 |
| 6 | 1 | 0.0150 |
| 6 | 2 | 0.0450 |
| 6 | 3 | 0.0900 |
| 6 | 4 | 0.1350 |
| 6 | 5 | 0.1620 |
| 6 | 6 | 0.1620 |
| 7 | 0 | 0.0009 |
| 7 | 1 | 0.0063 |
| 7 | 2 | 0.0220 |
| 7 | 3 | 0.0515 |
| 7 | 4 | 0.0903 |
| 7 | 5 | 0.1264 |
| 7 | 6 | 0.1475 |
| 7 | 7 | 0.1475 |
Poisson Distribution Formula
The Poisson distribution formula is:
P(X = k) = (e^-λ × λ^k) / k!Where:
- P(X = k) is the probability of k events occurring
- e is Euler’s number (≈ 2.71828)
- λ (lambda) is the average number of events per interval
- k is the number of events we’re calculating the probability for
How Do You Calculate the Poisson Distribution?
Determine λ (average rate of events)
Choose k (number of events you’re interested in)
Apply the formula
Simplify and calculate the result
Calculating P(X = 3) with λ = 2
- λ = 2
- k = 3
- P(X = 3) = (e^-2 × 2^3) / 3!
- P(X = 3) ≈ 0.180 or 18.0%
Calculate the probability of observing exactly 5 events, given an average rate of 3 events per interval, using the formula:
P(X = 5) = (e^(-3) * 3^5) / 5! = 0.1236
The probability of observing exactly 5 events, given an average rate of 3 events per interval, is approximately 0.1236 or 12.36%.
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