Poisson Distribution
Probability of k events in a fixed interval.
Formula
Variables
Example
If a call centre averages λ = 3 calls/min, P(0 calls in a minute) = e⁻³ ≈ 0.05.
How it works
The Poisson formula gives the chance of seeing exactly k events in a fixed interval when events occur independently and at a constant average rate λ. It comes from a counting model where events are rare relative to the interval length, so the only thing that matters is the average rate, not the exact timing of past events. The factor λ^k / k! counts how many ways k arrivals can occur, while e^{-λ} normalizes the probabilities so all possible counts add up to 1.
When to use it
Use it for count data over time, area, volume, or distance when events happen randomly and independently, such as defects per metre, arrivals per hour, or mutations per DNA segment. It is especially useful when you know the average rate and want the probability of an exact number of occurrences.
Worked example, step by step
Suppose a webpage receives an average of λ = 2.5 visits per minute. Find the probability of exactly k = 4 visits in one minute. Use P(X = k) = (λ^k e^{-λ}) / k! P(X = 4) = (2.5^4 e^{-2.5}) / 4! 2.5^4 = 39.0625 4! = 24 P(X = 4) = (39.0625 × e^{-2.5}) / 24 e^{-2.5} ≈ 0.082085 P(X = 4) ≈ (39.0625 × 0.082085) / 24 P(X = 4) ≈ 3.2064 / 24 P(X = 4) ≈ 0.1336 Final answer: P(X = 4) ≈ 0.134
Common mistakes to avoid
- Using the Poisson formula for non-count data, such as heights or temperatures
- Forgetting to use k! in the denominator, especially when k is larger than 1
- Mixing up the average rate λ with the observed count k or using the wrong time interval
Did You Know?
The Poisson distribution was first used to model the number of Prussian soldiers killed by horse kicks per year.
Share this formula
More in Statistics & Probability
View allArithmetic Mean
BasicThe average of a data set.
Median
BasicThe middle value separating the higher and lower halves.
Variance
IntermediateAverage squared deviation from the mean.
Standard Deviation
IntermediateSquare root of the variance; typical spread.
Z-Score
IntermediateNumber of standard deviations a value is from the mean.
Bayes' Theorem
IntermediateUpdates a probability given new evidence.