Formula & Calculator
Poisson Probability
Calculates the probability of observing exactly k events in a fixed interval, given a known average event rate.
Interpretation
P(X=k) = (λᵏ e⁻λ)/k!. Probability of k events in a fixed interval, with average rate λ. Used for rare events, call centres, and traffic flow.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| P(X=k) | Probability of exactly k events | |
| λ | Average rate of events | |
| k | Number of events observed |
What it means
The Poisson distribution models the number of events occurring in a fixed interval of time or space, given the average number of events λ. It assumes events are independent and occur at a constant rate. It is used in queuing theory, insurance claim modelling, traffic flow, and epidemiology (rare disease counts). The formula uses the exponential and factorial. The mean and variance both equal λ. Understanding the Poisson distribution is fundamental for modelling count data and for analysing rare events. It is also used in reliability (number of failures) and in biology (number of mutations).
Worked example
Poisson Probability – Two Examples
Real‑World| Parameter | Value |
|---|---|
| λ | 3 |
| k | 2 |
| Parameter | Value |
|---|---|
| λ | 5 |
| k | 5 |
Common mistakes
- Poisson distribution: Models the number of events in a fixed interval of time or space.
- Parameter λ: The average number of events – must be >0.
- k: Non‑negative integer (0,1,2,…).
- e: Euler’s number (~2.718) – use correct value.
- Independence: Assumes events occur independently with a constant rate.
Applications
The Poisson probability formula gives the probability of exactly k events occurring in a fixed interval of time or space, given a known average rate λ. It models rare events and is used in reliability (number of failures), quality control (defects per unit), and queueing theory (arrivals). Engineers use it to predict the number of defects in manufactured goods, to design spare parts inventory, and to assess the probability of extreme events. In biology, it models mutation rates. By using the Poisson distribution, professionals can quantify the likelihood of various counts, set service levels, and plan capacity. Its simplicity and wide applicability make it a staple in statistical practice.
- Quality control – number of defects per product or per batch
- Reliability engineering – number of failures over time
- Queueing theory – number of arrivals or service completions
- Inventory management – demand during lead time
- Biology and ecology – organism counts in a given area
Frequently Asked Questions
P(X=k) = (λᵏ · e⁻λ) / k!. It models the probability of exactly k events occurring in a fixed interval of time or space, given that events occur at a constant average rate λ and independently.
- Events occur independently.
- The average rate (λ) is constant.
- Two events cannot occur at exactly the same instant (for time).
- The probability of an event in a small interval is proportional to the interval length.
Both the mean and variance are equal to λ. This is a key property: E[X] = Var(X) = λ.
It is used to model count data: number of phone calls per hour, number of accidents per day, number of defective items in a batch, or number of customers entering a shop.
The Poisson can be derived as a limiting case of the binomial when n → ∞ and p → 0 such that np = λ. It is often used as an approximation for rare events.
P(X=0) = e⁻λ. This is the probability that no events occur in the interval.
Sum the probabilities from 0 to k: Σ_{i=0}^{k} (λⁱ e⁻λ) / i!. This is often looked up in tables or computed using software.
If the number of events in a fixed interval is Poisson(λ), then the inter‑arrival times are exponentially distributed with rate λ. They are complementary.
It models the arrival process of customers in a queue. The number of arrivals in a time period is often Poisson, and the service times may be exponential.
A Poisson process is a counting process where events occur continuously and independently at a constant average rate. It is a fundamental model in stochastic processes and has applications in telecommunications, traffic, and reliability.