Poisson Distribution Formula
Calculate the probability of a given number of events in a fixed interval.
Models rare events like calls, defects, or arrivals.
The Formula
P(X = k) = (λᵏ × e⁻λ) / k!
The Poisson distribution models the probability of a certain number of events occurring in a fixed time or space interval. It works best when events are independent and occur at a known average rate.
Variables
| Symbol | Meaning |
|---|---|
| P(X = k) | Probability of exactly k events occurring |
| λ | Average rate of events per interval (lambda) |
| k | Number of events you want the probability for |
| e | Euler's number (approximately 2.71828) |
| k! | k factorial |
Example 1
A call center averages 4 calls per minute. What is P(exactly 6 calls)?
λ = 4, k = 6
P(6) = (4⁶ × e⁻⁴) / 6! = (4096 × 0.01832) / 720
P(6) ≈ 0.1042 ≈ 10.4%
Example 2
A website gets 2 errors per hour on average. What is P(zero errors in an hour)?
λ = 2, k = 0
P(0) = (2⁰ × e⁻²) / 0! = (1 × 0.1353) / 1
P(0) ≈ 0.1353 ≈ 13.5%
When to Use It
Use the Poisson distribution when:
- Modeling the number of events in a fixed time period (calls, arrivals, emails)
- Counting defects per unit in manufacturing quality control
- Predicting traffic accidents, server failures, or rare diseases
- Events are independent and occur at a constant average rate
Key Notes
- Formula: P(k; λ) = λᵏ e⁻λ / k!: λ (lambda) is the average number of events in the interval. k is the number of events you want the probability for. e ≈ 2.71828.
- Mean equals variance: For a Poisson distribution, both the mean and the variance equal λ. If observed variance is much larger than the mean in real data, consider a negative binomial model instead.
- Three conditions must hold: Events must occur independently, at a constant average rate, and two events cannot occur at exactly the same instant. If these fail, Poisson is not appropriate.
- Approximates binomial for large n, small p: When n ≥ 20 and p ≤ 0.05, the binomial B(n, p) is well-approximated by Poisson with λ = np. This simplifies calculations significantly.
- Common real-world applications: Call center arrivals per minute, number of defects in a production run, radioactive decay events per second, and traffic accidents per month all follow Poisson distributions.