Poisson Distribution
The Poisson distribution models the number of events occurring in a fixed interval of time, space, or other continuous domain, given that events occur independently at a constant average rate. It is widely used in queueing theory, reliability engineering, and epidemiology. The distribution is uniquely characterized by its single parameter λ, which represents both the mean and the variance.
Formula
P(X = k) = (λ^k · e^(-λ)) / k!, for k = 0, 1, 2, ...
Mean (Expected Value)
λ
Variance
λ
Parameters
The average number of events per interval. Must be positive (λ > 0). Also called the rate parameter or intensity.
Key Properties
- •The mean equals the variance: E(X) = Var(X) = λ
- •Events must occur independently of each other
- •The rate of events is constant across the interval
- •The sum of independent Poisson random variables is also Poisson: if X ~ Poi(λ₁) and Y ~ Poi(λ₂), then X + Y ~ Poi(λ₁ + λ₂)
- •For large λ (typically λ ≥ 10), the Poisson distribution is well-approximated by N(λ, λ)
- •The Poisson distribution is the limit of the binomial distribution as n → ∞ and p → 0 with np = λ
Example
A call center receives an average of 8 calls per hour. What is the probability that exactly 5 calls arrive in a given hour?
Here λ = 8 and k = 5. P(X = 5) = (8^5 · e^(-8)) / 5! = (32768 · 0.000335) / 120 = 10.98 / 120 ≈ 0.0916.
Result: P(X = 5) ≈ 0.0916, or about 9.16%
There is about a 9.16% chance of receiving exactly 5 calls in a given hour when the average rate is 8. Since 5 is below the mean of 8, this is a somewhat below-average hour but not rare.
When to Use
- ✓When counting the number of rare or random events in a fixed interval (calls per hour, defects per unit, accidents per month)
- ✓When events occur independently and the average rate is constant over time or space
- ✓As an approximation to the binomial distribution when n is large and p is small (n ≥ 20, p ≤ 0.05)
- ✓When modeling arrival processes in queueing theory or demand for services
Common Mistakes
- ✗Using Poisson when events are not independent or the rate is not constant. For example, website traffic with peak hours violates the constant rate assumption.
- ✗Confusing the rate for different interval lengths. If λ = 8 per hour, then for 30 minutes use λ = 4, not 8.
- ✗Forgetting that the Poisson distribution has no upper bound on k. While P(X = k) gets very small for large k, it never reaches zero.
- ✗Assuming Poisson is appropriate when the variance is much larger or smaller than the mean (overdispersion or underdispersion). The Poisson requires mean ≈ variance.
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Common questions about Poisson Distribution
Use Poisson when you are counting events in a continuous interval (time, area, volume) where there is no fixed upper limit on the count, and events occur at a constant average rate. Use binomial when you have a fixed number of discrete trials with two outcomes each. As a practical bridge, the Poisson approximation to the binomial is useful when n > 20 and p < 0.05: set λ = np and use the Poisson PMF for easier calculations.
The property E(X) = Var(X) = λ is called equidispersion. It means the spread of the data is entirely determined by the average rate. If you observe data where the sample variance is close to the sample mean, a Poisson model may be appropriate. If the variance is much larger than the mean (overdispersion), consider the negative binomial distribution instead. This mean-variance relationship is also a quick diagnostic tool to check whether Poisson is a reasonable model for your data.
If events occur at rate λ per unit time, then in t time units the number of events follows a Poisson distribution with parameter λt. For example, if a store averages 12 customers per hour (λ = 12/hour), then in 20 minutes (t = 1/3 hour), use λ′ = 12 × (1/3) = 4. In 2 hours, use λ′ = 12 × 2 = 24. Always scale the rate parameter to match the interval you are analyzing.