The Poisson Distribution
Statistics & Probability Guide
The Poisson Distribution: Complete Guide
Everything you need to understand the Poisson distribution — formula, assumptions, worked examples, real-world applications, and how it compares to the Binomial — from first principles to Poisson regression.
Table of Contents
Foundations
What Is the Poisson Distribution? Definition and Origins
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval — of time, area, length, or volume — when those events happen independently and at a known, constant average rate. It is one of the cornerstones of modern probability theory, and its fingerprint is everywhere: in call centers and hospital admissions, in particle physics and genome sequencing, in insurance actuarial tables and web server logs. If you are counting rare, independent events in a continuous medium, the Poisson distribution is almost certainly the right starting point.
The distribution is named after Siméon Denis Poisson, a French mathematician and physicist at the École Polytechnique in Paris who introduced it in his 1837 treatise Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Poisson derived the distribution as a limiting case of the Binomial distribution when the number of trials is very large and the probability of success per trial is very small.
λ
Lambda — the only parameter. Controls both the mean AND the variance of the distribution
1837
Year Siméon Denis Poisson formally introduced the distribution in Paris
∞
Theoretical support: k can be any non-negative integer — 0, 1, 2, 3 … with no upper bound
What makes the Poisson distribution particularly remarkable is its parsimony: it is fully defined by a single parameter, λ (lambda), which simultaneously serves as the mean and the variance of the distribution. No other common distribution shares this property. This one-parameter elegance makes it computationally tractable and statistically convenient — but it also imposes a strong constraint: if your data shows a variance meaningfully different from its mean, the Poisson model is already under stress. We will come back to that problem — called overdispersion — in the section on limitations.
The Intuition Behind Poisson — Before the Formula
Before diving into algebra, it is worth building intuition. Imagine a stretch of highway where accidents occur at an average rate of 2 per week, independently of each other. You want to know: what is the probability of exactly 5 accidents next week? The accidents are discrete (countable), they occur in a fixed interval (one week), the average rate is known and constant (2 per week), and they are independent (one accident does not cause another). This is precisely the setup the Poisson distribution was designed for.
The key insight is this: the Poisson distribution is not about a fixed number of trials with a success probability (that would be Binomial). It is about counting events in a continuous medium — time, space, or any measurable substrate — where the expected density of events is known. The formula then tells you the entire probability distribution for how many events you will actually observe.
Who Uses the Poisson Distribution?
Google and Amazon Web Services (AWS) use Poisson-based models to predict server request rates and scale infrastructure. Lloyd’s of London and major US insurers use Poisson processes to model rare catastrophic events in actuarial models. The US Centers for Disease Control and Prevention (CDC) applies Poisson regression to model disease incidence rates. CERN uses Poisson statistics to model radioactive particle detection events. And virtually every queuing system — from McDonald’s drive-throughs to United Airlines check-in counters — is designed using Poisson arrival assumptions.
The Mathematics
The Poisson Distribution Formula — The PMF Explained
The Poisson distribution is defined by its probability mass function (PMF), which gives the probability of observing exactly k events in a fixed interval, given that the average rate of events is λ. This is the equation you need to understand at the level of every component — not just as a formula to memorise but as a model to reason with.
Poisson Probability Mass Function (PMF)
P(X = k) = (λk · e−λ) / k!
Where: X = discrete random variable (number of events), k = 0, 1, 2, 3, … (non-negative integer), λ = average rate of events per interval (lambda > 0), e = Euler’s number ≈ 2.71828, k! = k factorial = k × (k−1) × … × 2 × 1 (with 0! = 1 by convention)
Every component of this formula carries specific probabilistic meaning. The numerator has two parts: λ^k grows as you consider larger counts — it reflects the fact that higher expected rates make higher observed counts more likely. The term e^−λ is a normalisation factor that ensures probabilities across all possible k values sum exactly to 1. The denominator k! accounts for the combinatorial explosion — as k grows, this term scales the probability accordingly.
Deriving the Poisson from the Binomial
Poisson’s own derivation proceeds as follows. Consider a Binomial distribution with n trials and success probability p = λ/n. As n → ∞ and p → 0 such that np = λ remains constant, the Binomial PMF converges to the Poisson PMF. This is the “law of rare events.” The practical implication: when n > 100 and p < 0.01, with np = λ, the Poisson approximation to the Binomial is accurate and computationally simpler.
The CDF — Cumulative Poisson Probabilities
The Poisson cumulative distribution function (CDF) gives the probability of observing at most k events: P(X ≤ k) = Σ (from j=0 to k) of P(X = j). In R: ppois(k, lambda). In Python: scipy.stats.poisson.cdf(k, mu=lambda). In Excel: =POISSON.DIST(k, lambda, TRUE).
Quick Exam Tip: When P(X ≥ k)
Many exam problems ask for “at least k events.” Remember: P(X ≥ k) = 1 − P(X ≤ k−1). This complementary approach saves significant computation when k is small — for example, “at least 1 event” = 1 − P(X = 0) = 1 − e^−λ, often the fastest calculation in the distribution.
The Moment Generating Function (MGF)
The moment generating function of the Poisson distribution is M(t) = e^(λ(e^t − 1)). From this MGF, all moments can be derived by differentiating with respect to t and evaluating at t = 0. The first derivative at t = 0 gives E[X] = λ. The second derivative gives E[X²] = λ² + λ, leading to Var(X) = λ. This derivation confirms the defining property: mean = variance = λ.
Model Conditions
The Four Assumptions of the Poisson Distribution
No model is universally valid — and the Poisson distribution is no exception. Its mathematical elegance rests on four specific assumptions that must hold (at least approximately) for the model to be meaningful. Knowing these assumptions is not just exam preparation: in applied statistics, checking whether your data satisfies them is the first thing any competent analyst does before fitting a Poisson model.
Assumption 1: Independence of Events
Events must be independent — the occurrence of one event does not change the probability of any other event occurring. In a call center model, this means one phone call does not cause or prevent another. Independence is the most fundamental assumption and the most commonly violated one in practice. When events cluster — because infected individuals transmit disease, or because a web server attack generates cascading failures — independence fails, and the Poisson distribution underestimates the true variance of the data.
Assumption 2: Constant Rate (Stationarity)
The average rate λ must be constant across the interval being modelled. If you are modelling hospital emergency admissions per hour but rates spike during flu season and plummet in summer, a single λ is insufficient — your data is non-stationary. In practice, analysts often handle non-stationarity by segmenting data or using Poisson regression with covariates.
Assumption 3: Events Cannot Occur Simultaneously
Two events cannot happen at the exact same instant in continuous time. In practice, this is usually satisfied automatically when your interval is sufficiently fine-grained. This assumption is sometimes stated as: in any infinitesimally small interval dt, the probability of one event is λdt and the probability of two or more events is negligible (o(dt)).
Assumption 4: Proportionality in Small Intervals
The probability of an event in a small interval is proportional to the length of that interval. If the rate is λ per unit time, the probability of an event in a small interval of length Δt is approximately λΔt. This assumption connects directly to the exponential distribution for inter-event times.
When Assumptions Are Violated — Overdispersion: The most common real-world problem with Poisson models is overdispersion — when the observed variance exceeds the mean (Var(X) > E[X]). In these cases, the Negative Binomial distribution or a Zero-Inflated Poisson (ZIP) model provides a better fit. Always compute the dispersion ratio (variance/mean) before committing to a Poisson model for your data.
Testing Poisson Goodness-of-Fit
In practice, you do not just assume the Poisson fits — you test it. The standard approach uses a chi-square goodness-of-fit test comparing observed frequencies of each count (0, 1, 2, …) with expected frequencies under a Poisson model with estimated λ. If the test statistic is statistically significant, the Poisson model is rejected for that dataset.
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Mean, Variance, and Shape of the Poisson Distribution
The Poisson distribution has a set of mathematical properties that are unusually clean and useful. The most celebrated is the mean-variance equality: both the expected value (mean) and the variance of a Poisson distribution equal λ.
Poisson Distribution — Key Properties
E[X] = λ | Var(X) = λ | SD(X) = √λ
Skewness: 1/√λ | Kurtosis (excess): 1/λ | Mode: ⌊λ⌋ (floor of λ) when λ is not an integer; both ⌊λ⌋ and λ when λ is an integer
Why Mean = Variance — The Diagnostic Implication
The equality E[X] = Var(X) = λ gives you a powerful diagnostic tool: if you collect count data and compute the sample mean and sample variance, a Poisson model is plausible only if these two statistics are approximately equal. The ratio Var(X)/E[X] should be close to 1.0 for Poisson; significantly above 1.0 signals overdispersion; significantly below 1.0 signals underdispersion.
Shape: How λ Controls the Distribution’s Appearance
When λ is small (say λ = 0.5 or λ = 1), the distribution is heavily right-skewed — most probability mass sits at 0 and 1. As λ increases toward 5 or 10, the distribution becomes more symmetric with a discernible mode at k ≈ λ. At λ ≥ 30, the distribution is nearly indistinguishable from a Normal distribution by visual inspection.
The Additive Property — Superposition of Poisson Processes
One of the most practically useful properties: if X₁ ~ Poisson(λ₁) and X₂ ~ Poisson(λ₂) are independent, then X₁ + X₂ ~ Poisson(λ₁ + λ₂). The sum of independent Poisson random variables is also Poisson, with rate equal to the sum of the individual rates.
Normal Approximation to the Poisson
When λ ≥ 10 (conservatively λ ≥ 30), the Poisson distribution is well approximated by a Normal distribution: X ~ Poisson(λ) ≈ N(λ, λ). A continuity correction improves the approximation: P(X ≤ k) ≈ P(Z ≤ (k + 0.5 − λ)/√λ), where Z is a standard Normal random variable.
Solved Problems
Worked Examples — Poisson Distribution Step by Step
The best way to internalise the Poisson distribution is to work through concrete, numerical problems. This section presents five fully solved examples spanning different real-world contexts — from call centers to radioactive decay — with every calculation step shown.
Example 1: Call Center Arrivals
A customer service center receives an average of 3 calls per minute. What is the probability of receiving exactly 5 calls in a given minute?
Given: λ = 3, k = 5
P(X = 5) = (3⁵ × e⁻³) / 5!
= (243 × 0.04979) / 120
= 12.09 / 120 ≈ 0.1008
There is approximately a 10.08% probability of receiving exactly 5 calls in a given minute when the average is 3.
Example 2: Manufacturing Defects
A factory produces fabric with an average of 1.5 defects per square meter. What is the probability that a randomly selected square meter has no defects?
Given: λ = 1.5, k = 0
P(X = 0) = (1.5⁰ × e⁻¹·⁵) / 0!
= (1 × 0.2231) / 1
= 0.2231
There is a 22.31% probability of finding a defect-free square meter. This equals e^−λ = e^−1.5 — the useful shortcut for k = 0.
Example 3: Hospital Emergency Admissions
An emergency room admits an average of 4 patients per hour. What is the probability of admitting at least 2 patients in any given hour?
Given: λ = 4. Use complement: P(X ≥ 2) = 1 − P(X = 0) − P(X = 1)
P(X = 0) = e⁻⁴ = 0.01832
P(X = 1) = (4 × e⁻⁴) / 1 = 4 × 0.01832 = 0.07326
P(X ≥ 2) = 1 − 0.01832 − 0.07326 = 0.9084
There is a 90.84% probability that at least 2 patients arrive in a given hour.
Example 4: Radioactive Decay (Physics)
A radioactive isotope emits particles at an average rate of 2.5 per second. What is the probability of observing exactly 4 particles in a given second?
Given: λ = 2.5, k = 4
P(X = 4) = (2.5⁴ × e⁻²·⁵) / 4!
= (39.0625 × 0.08208) / 24
= 3.2063 / 24 ≈ 0.1336
The probability of observing exactly 4 particles in one second is approximately 13.36%.
Example 5: Changing the Interval Length
Website traffic arrives at an average rate of 6 visitors per minute. What is the probability of receiving exactly 3 visitors in a 30-second interval?
Key Step: Rescale λ to the new interval. λ per 30 seconds = 6 × (30/60) = 3. Now k = 3, λ = 3.
P(X = 3) = (3³ × e⁻³) / 3!
= (27 × 0.04979) / 6
= 1.3442 / 6 ≈ 0.2240
The probability is approximately 22.40%. Always rescale λ proportionally when the interval in your problem differs from the interval in which λ was originally expressed.
Distribution Comparisons
Poisson vs. Binomial: When to Use Which?
One of the most frequently tested concepts in probability courses is knowing when to use the Poisson distribution versus the Binomial distribution. Both model discrete counts, but they arise from fundamentally different setups — and applying the wrong one is a common source of errors.
Binomial Distribution — Use When:
- Fixed number of trials n is known and finite
- Each trial has exactly two outcomes: success or failure
- Probability of success p is constant and moderate (not tiny)
- Trials are independent of each other
- You are counting the number of successes in n trials
- Example: Number of heads in 20 coin flips; defective items in a sample of 50
Poisson Distribution — Use When:
- Events occur in a continuous interval (time, space, area)
- The average rate λ per interval is known; n and p are not
- Events are rare relative to the opportunities (small p, large n)
- Events are independent and cannot occur simultaneously
- You are counting events per fixed interval, not per fixed trials
- Example: Calls per minute; accidents per week; defects per square meter
The Poisson Approximation Rule
When n ≥ 50 and p ≤ 0.1 (with λ = np reasonably small, typically λ ≤ 10), the Binomial(n, p) distribution is well approximated by Poisson(λ = np). This is why the Poisson is sometimes called the “law of rare events.”
| Feature | Binomial Distribution | Poisson Distribution |
|---|---|---|
| Parameters | n (trials) and p (success probability) | λ (average rate) only |
| Number of trials | Fixed, finite n | Not fixed; events in continuous interval |
| Range of k | 0, 1, 2, …, n (bounded above by n) | 0, 1, 2, 3, … (unbounded) |
| Mean | np | λ |
| Variance | np(1−p) | λ (always equals mean) |
| Mean = Variance? | Only when p = 0 (trivial) | Always, by definition |
| Typical use case | Quality sampling, clinical trials, survey responses | Arrival processes, rare events, count data per interval |
| Software PMF command (R) | dbinom(k, n, p) | dpois(k, lambda) |
Stochastic Processes
The Poisson Process — The Theory Behind the Distribution
The Poisson process is the continuous-time stochastic process from which the Poisson distribution is derived. It underlies telephone switching networks, packet routing on the internet, queuing at retail counters, insurance claims processing, and radioactive decay monitoring. Understanding the Poisson process provides a deeper, more unified view of what the distribution is actually modelling.
Formal Definition
A counting process {N(t), t ≥ 0} is a homogeneous Poisson process with rate λ if: (1) N(0) = 0; (2) it has independent increments; (3) it has stationary increments; and (4) N(t) ~ Poisson(λt) for any t > 0. The number of events in any interval of length t follows Poisson with mean λt — which is why rescaling λ by the interval length is mathematically justified.
Inter-Arrival Times Follow the Exponential Distribution
One of the most elegant results in probability theory: if events arrive according to a Poisson process with rate λ, the times between consecutive events follow an Exponential distribution with parameter λ (mean 1/λ). The memoryless property of the Exponential distribution directly corresponds to the independence and stationarity of the Poisson process.
Non-Homogeneous Poisson Process
Real-world event rates are not always constant. A non-homogeneous Poisson process allows the rate to vary over time: λ(t) becomes a function of time. The number of events in an interval [s, t] follows Poisson with mean ∫ λ(u) du from s to t. This connects directly to Poisson regression and Generalized Linear Models (GLMs).
Compound Poisson Process
A compound Poisson process models the cumulative sum of random “jump sizes” at Poisson-distributed event times. If each event carries a random cost (like an insurance claim of random severity), the total accumulated cost is a compound Poisson random variable — the foundation of collective risk theory in actuarial science.
Applied Statistics
Real-World Applications of the Poisson Distribution
The Poisson distribution would not have survived 185+ years of active use if it were merely a textbook curiosity. This section surveys the major application domains across key US and UK industries.
Telecommunications and Computer Networks
Danish mathematician Agner Krarup Erlang at the Copenhagen Telephone Company applied Poisson models to telephone traffic in the early 20th century, deriving the Erlang-B and Erlang-C formulas that still govern network capacity planning. Today, engineers at Verizon, AT&T, and British Telecom (BT) use Poisson-based queuing models to determine capacity needed to meet service level agreements.
Epidemiology and Public Health
The US CDC, the UK Health Security Agency (UKHSA), and academic schools of public health at Johns Hopkins Bloomberg School of Public Health and London School of Hygiene and Tropical Medicine (LSHTM) routinely apply Poisson regression to model disease incidence rates. During COVID-19, Poisson models were used by Imperial College London‘s MRC Centre to estimate baseline excess mortality rates.
Manufacturing and Quality Control
Quality control engineers at Boeing, Ford, and Rolls-Royce use the Poisson distribution to model defects per unit area or per production run. The ISO and the American Society for Quality (ASQ) both reference Poisson-based control chart methodology in manufacturing quality standards.
Finance and Insurance — Actuarial Science
Actuaries at Lloyd’s of London, Swiss Re, and Munich Re model catastrophic event frequencies using Poisson processes. Under the Basel III framework, the probability of a given number of counterparty defaults is sometimes modelled using Poisson or compound Poisson distributions.
Ecology and Biology
Field ecologists use the Poisson distribution to model spatial patterns of organisms. When organisms are randomly distributed, count per equal-area quadrat follows Poisson. In molecular biology, the number of mutations per genome under neutral evolution is modelled as Poisson — a foundational concept in phylogenetics and molecular clock models.
Traffic Engineering and Transportation
Traffic engineers at the US Federal Highway Administration (FHWA) and Transport for London (TfL) use Poisson distributions to model vehicle arrivals at intersections. The Highway Safety Manual explicitly recommends Poisson and Negative Binomial models as the standard for safety performance functions.
“The Poisson distribution is one of the most powerful and pervasive models in applied statistics, not because nature is simple, but because independence and constant rates are surprisingly good approximations to a remarkable variety of real phenomena — from the sub-atomic to the cosmic.” — Adapted from Durrett, Probability: Theory and Examples, Cambridge University Press.
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Poisson Regression — Modelling Count Data with Covariates
The Poisson distribution is not just a standalone probability model — it is the foundation of Poisson regression, one of the most important models in applied statistics for count data. When you want to understand what factors predict the number of events, Poisson regression allows λ to vary as a function of predictor variables. It is a Generalized Linear Model (GLM) with a Poisson family and a log link function.
The Poisson Regression Model
The model specifies: log(E[Y]) = β₀ + β₁X₁ + β₂X₂ + … + βₙXₙ. The log link function ensures the predicted mean count is always positive. The exponentiated coefficients — e^βᵢ — are incidence rate ratios (IRRs): they give the multiplicative change in the expected count for a one-unit increase in Xᵢ, holding other variables constant. The model is estimated by maximum likelihood estimation (MLE).
Offset Terms in Poisson Regression
When count data comes from different-sized intervals — different population sizes, different time periods — a rate model is needed. This is handled by including an offset term: log(E[Y]) = log(exposure) + β₀ + β₁X₁ + …. The log(exposure) term accounts for varying denominators without estimating a separate coefficient. This technique is ubiquitous in epidemiology.
When Poisson Regression Fails — Overdispersion and Zero Inflation
Poisson regression requires that the conditional variance equals the conditional mean. When real data shows overdispersion, the standard solution is Negative Binomial regression. When the data also has an excess of zeros, Zero-Inflated Poisson (ZIP) or Zero-Inflated Negative Binomial (ZINB) models are used. These are standard in applied statistics courses at UCLA, Duke, Manchester, and Nottingham.
Step-by-Step Guide
How to Solve Any Poisson Distribution Problem
Solving Poisson distribution problems follows a consistent pattern regardless of the application domain. Internalise this seven-step process and you will be able to tackle any Poisson problem systematically.
1
Identify the Random Variable and the Count
Read the problem carefully. Identify what is being counted (X), over what interval, and what specific count (k) you need the probability for. State explicitly: X = number of [events] per [interval]. Is X plausibly discrete and unbounded? Does the setup involve counting rare, independent events in a continuous medium? If yes — Poisson is likely the right model.
2
Extract or Calculate λ for the Given Interval
Find the average rate of events per interval. If the problem gives a rate for a different interval length, rescale proportionally: λ_new = λ_given × (new interval / given interval). Always work in consistent units. This rescaling step is where most errors occur — be disciplined about it.
3
Check Which Probability Is Needed
Is the question asking for P(X = k), P(X ≤ k), P(X ≥ k), or P(a ≤ X ≤ b)? Choose the most efficient computation route. For P(X ≥ 1), always use 1 − P(X = 0) = 1 − e^−λ. For “at least k” with small k, use the complement rule.
4
Apply the Poisson PMF Formula
Substitute λ and k into P(X = k) = (λ^k × e^−λ) / k!. Compute each component step by step: λ^k first, then e^−λ (use e ≈ 2.71828), then k!. Multiply numerator, then divide by denominator. Show every step for full marks in assignments.
5
Sum for Cumulative or Range Probabilities
For P(X ≤ k), sum P(X = 0) + P(X = 1) + … + P(X = k). Use Poisson tables if available. In software: ppois(k, lambda) in R or scipy.stats.poisson.cdf(k, mu=lambda) in Python.
6
Check Using the Mean and Variance
Quick sanity checks: E[X] = Var(X) = λ. The most probable outcome (mode) is approximately ⌊λ⌋. The distribution should tail off quickly for k >> λ. If your computed probability for k much larger than λ is suspiciously large, recheck your calculation.
7
Interpret the Result in Context
Always state your answer in the context of the problem: “There is a 10.08% probability of receiving exactly 5 calls in a given minute.” Never just write “0.1008” without interpretation. Contextual interpretation is worth marks — and it is the difference between computation and statistical thinking.
Poisson in Software — Key Commands: R:
dpois(k, lambda) for PMF, ppois(k, lambda) for CDF (P(X ≤ k)), qpois(p, lambda) for quantiles, rpois(n, lambda) for random samples. Python (SciPy): scipy.stats.poisson.pmf(k, mu=lambda), scipy.stats.poisson.cdf(k, mu=lambda). Excel: =POISSON.DIST(k, lambda, FALSE) for PMF; =POISSON.DIST(k, lambda, TRUE) for CDF.
Critical Analysis
Limitations of the Poisson Distribution and When to Use Alternatives
No model is universally appropriate — and part of statistical maturity is knowing when a model breaks down. The Poisson distribution is a powerful default for count data, but it has genuine limitations that appear regularly in applied work.
Overdispersion — Variance > Mean
Overdispersion is by far the most common Poisson violation in real data. Common causes include unobserved heterogeneity, positive event correlation, and zero excess. The standard diagnostic is the dispersion ratio: variance/mean >> 1 is a red flag. The standard remedies are Negative Binomial regression (parametric fix) or quasi-Poisson with robust standard errors (semi-parametric fix).
Zero Inflation
Zero inflation occurs when the data contains more zeros than the Poisson model predicts. This is common in healthcare utilisation data, ecological abundance surveys, and count data with structural zeros. Zero-Inflated Poisson (ZIP) models use a mixture of a point mass at zero and a Poisson distribution. In R, fitting ZIP models uses the pscl package’s zeroinfl() function.
Spatial and Temporal Dependence
The Poisson distribution assumes independence, but spatial and temporal autocorrelation violate this. Disease cases cluster geographically; accidents cluster temporally after bad weather. Spatial Poisson regression with random effects, Bayesian hierarchical Poisson models, and autoregressive count time series models are the modern solutions.
When the Rate Is Not Constant
If events occur at different rates at different times, solutions include: segmenting data into homogeneous time periods; using a non-homogeneous Poisson process with parametric λ(t); or incorporating time as a covariate in Poisson regression.
Quick Reference: Choosing the Right Count Model
Variance ≈ Mean, no excess zeros? → Standard Poisson
Variance > Mean, continuous overdispersion? → Negative Binomial
Excess zeros + Poisson for non-zeros? → Zero-Inflated Poisson (ZIP)
Excess zeros + overdispersion? → Zero-Inflated Negative Binomial (ZINB)
Correlated/clustered counts? → Random effects Poisson or Bayesian hierarchical model
Large λ (≥ 10–30)? → Normal approximation with mean λ and variance λ may suffice
Frequently Asked Questions
Frequently Asked Questions About the Poisson Distribution
What is the Poisson distribution?
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval — of time, area, or space — when events happen independently at a constant average rate λ. Named after French mathematician Siméon Denis Poisson, who introduced it in 1837, the distribution is defined by a single parameter λ (lambda), which equals both the mean and the variance. It is used in statistics, engineering, biology, finance, and public health to model count data — particularly rare or infrequent events such as customer arrivals, disease cases, defects per unit, or radioactive decay events.
What is the formula for the Poisson distribution?
The Poisson probability mass function (PMF) is: P(X = k) = (λ^k × e^−λ) / k!, where k is the number of events (a non-negative integer: 0, 1, 2, …), λ is the average event rate per interval (lambda > 0), and e ≈ 2.71828 (Euler’s number). For k = 0, the formula simplifies to P(X = 0) = e^−λ. The mean and variance are both equal to λ. The CDF (cumulative probability) is computed by summing the PMF values from 0 to k.
What are the assumptions of the Poisson distribution?
The four core Poisson assumptions are: (1) Independence — events occur independently of each other; (2) Constant rate — the average rate λ is constant across the interval; (3) Non-simultaneous — two events cannot occur at the exact same instant; (4) Proportionality — the probability of an event in a small interval is proportional to the interval’s length. When independence or constant rate assumptions are violated, the Negative Binomial, Zero-Inflated Poisson, or hierarchical models are typically more appropriate.
What is λ (lambda) in the Poisson distribution?
Lambda (λ) is the rate parameter — the average number of events expected in a given fixed interval. It is the only parameter needed to fully define the distribution, and it simultaneously equals the mean AND the variance: E[X] = Var(X) = λ. When the problem specifies a rate for a different time interval, always rescale: λ_new = λ_given × (new interval / given interval). Lambda can be any positive real number; it does not have to be an integer.
What is the difference between Poisson and Binomial distributions?
The Binomial distribution models the number of successes in a fixed, finite number of independent trials n, with constant success probability p. The Poisson models events in a continuous interval when rate λ is known but n and p are not. Key differences: Binomial has an upper bound of n on k; Poisson has no upper bound. Binomial variance = np(1−p); Poisson variance = λ = mean. Use Poisson as an approximation to Binomial when n ≥ 50 and p ≤ 0.1 (with λ = np).
When should you use the Poisson distribution?
Use the Poisson distribution when you are (1) counting discrete events in a fixed continuous interval; (2) the average rate per interval is known or estimable; (3) events are rare relative to opportunities; (4) events are independent; and (5) simultaneous events are impossible or negligible. Classic situations: calls per hour; accidents per week; defects per production run; disease cases per 100,000 population per year; bacteria colonies per petri dish. If your sample variance significantly exceeds the sample mean, consider the Negative Binomial distribution instead.
Why does the Poisson distribution have mean equal to variance?
The equality E[X] = Var(X) = λ is a direct mathematical consequence of the PMF’s structure, proven using the moment generating function M(t) = e^(λ(e^t − 1)). The practical implication: observed variance ≈ observed mean is the empirical signature of a Poisson process. If the observed variance significantly exceeds the mean, this signals clustering or dependence between events.
What is a Poisson process?
A Poisson process is a continuous-time stochastic process where events occur independently at a constant average rate λ per unit time. The number of events in any interval of length t follows Poisson(λt). The waiting time between consecutive events follows an Exponential distribution with rate λ (mean = 1/λ). Non-overlapping time intervals are independent. Applications include telephone call arrivals, server requests, radioactive particle emissions, and earthquakes above a threshold magnitude.
Is the Poisson distribution symmetric?
No — the Poisson distribution is right-skewed when λ is small, because it is bounded at zero on the left but has no upper bound on the right. Skewness equals 1/√λ, so smaller λ values produce more skewed distributions. As λ increases, skewness decreases and the distribution becomes more symmetric, approximating a Normal distribution by the Central Limit Theorem. For λ ≥ 10, the Normal approximation is reasonably accurate; for λ ≥ 30, it is excellent.
How do you calculate Poisson probabilities in Excel?
In Excel, use the POISSON.DIST function: =POISSON.DIST(k, lambda, cumulative). For the PMF (exactly k events): set cumulative = FALSE, e.g., =POISSON.DIST(5, 3, FALSE) gives P(X = 5) when λ = 3. For the CDF (at most k events): set cumulative = TRUE. For “at least k”: use =1 − POISSON.DIST(k−1, lambda, TRUE). In R: dpois(k, lambda) for PMF; ppois(k, lambda) for CDF. In Python: scipy.stats.poisson.pmf(k, mu=lambda) and scipy.stats.poisson.cdf(k, mu=lambda).
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