P(λ)

Poisson Distribution Calculator

PMF · CDF · Survival · Range Probability · Bar Chart

Compute P(X=k) = e−λλk/k!, cumulative P(X≤k), P(X>k), and range P(a≤X≤b) for any rate λ. Includes full probability table and canvas bar chart.

Quick Examples

Rate > 0
Integer ≥ 0

What Is the Poisson Distribution?

The Poisson distribution is a fundamental discrete probability distribution that models the number of times a random event occurs within a fixed interval of time, space, or another continuous medium — provided events happen independently and at a constant average rate λ (lambda). It was first derived by French mathematician Siméon Denis Poisson in 1837 as a limiting case of the binomial distribution when n is large and p is small.

Unlike the binomial distribution, which counts successes in a fixed number of trials, the Poisson distribution models count data where there is no theoretical upper bound. You can have 0, 1, 2, … any number of events in an interval. Classic scenarios include the number of customers arriving at a checkout in an hour, the number of emails received per day, or the number of flaws in a metre of fabric.

The PMF Formula and Log-Space Computation

The probability mass function (PMF) is:

P(X = k) = e−λ × λk / k!  for k = 0, 1, 2, …

For large λ or k, computing λk and k! directly causes arithmetic overflow. The solution is log-space computation:

log P(X = k) = −λ + k × ln(λ) − ln(k!) P(X = k) = exp(−λ + k × ln(λ) − ln(k!))

where ln(k!) is efficiently accumulated using a running sum: ln(k!) = Σi=1k ln(i). This approach is numerically stable for λ up to 200 and k up to 150.

Properties of the Poisson Distribution

PropertyValueNotes
Mean (μ)λExpected number of events
Variance (σ²)λEqual to mean — unique property
Std Dev (σ)√λSpread of the distribution
Skewness1/√λRight-skewed; symmetric for large λ
Kurtosis (excess)1/λApproaches normal as λ → ∞
Mode⌊λ⌋ and ⌊λ⌋−1 if λ integerMost probable count

The equal mean-variance property (μ = σ² = λ) is a defining characteristic. In practice, if your count data shows variance much greater than the mean (overdispersion), the negative binomial distribution may be more appropriate.

Additive Property and Superposition

If X ~ Poisson(λ1) and Y ~ Poisson(λ2) are independent, then X + Y ~ Poisson(λ1 + λ2). This additive property makes Poisson processes easy to combine — the total arrival rate of two independent Poisson streams is simply the sum of their rates.

Connection to the Exponential Distribution

If events follow a Poisson process with rate λ events per unit time, then the inter-arrival time between consecutive events follows an Exponential distribution with rate λ (mean 1/λ). This duality is the cornerstone of queuing theory and the M/M/1 queue model:

  • Arrivals: Customer count per hour ~ Poisson(λ)
  • Waiting time: Time until next customer ~ Exponential(λ)
  • Service time: If service rate is μ, time per service ~ Exponential(μ)

Poisson Approximation to Binomial

When n is large (≥ 20) and p is small (≤ 0.05), computing binomial probabilities becomes difficult. Since the mean is λ = np, we approximate Binomial(n, p) by Poisson(λ = np). The approximation error is bounded by min(p, λ/n) and improves as n → ∞ with λ fixed.

Real-World Applications

  • Queuing theory / call centers: Number of customer arrivals per time slot; M/M/1 queue analysis.
  • Telecommunications: Packet arrivals in network buffers; number of dropped calls per minute.
  • Epidemiology: Number of disease cases per district per year for rare conditions.
  • Radioactive decay: Geiger counter clicks per second follow Poisson perfectly.
  • Insurance: Number of claims per policyholder per year.
  • Website traffic: Page requests per second, especially during off-peak hours.
  • Manufacturing quality: Number of defects per unit area on a production line.
  • Ecology: Rare species sightings per hectare in a survey.
  • Astronomy: Photon counts per second arriving at a telescope detector.

Frequently Asked Questions

What is the Poisson distribution?
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval (time, space, etc.) when events happen independently at a constant average rate λ. The PMF is P(X=k) = e−λλk/k! for k = 0, 1, 2, … It is named after Siméon Denis Poisson (1837) and arises naturally as a limit of the binomial distribution.
What does lambda (λ) represent?
Lambda (λ) is the rate parameter — the expected number of events per interval. For example, if a call center receives an average of 5 calls per hour, λ = 5. Lambda must be a strictly positive real number. It simultaneously equals the mean and variance of the distribution, so λ can be estimated directly from data as the sample average count.
When should I use Poisson vs binomial distribution?
Use binomial when you have a fixed number n of independent trials, each with success probability p. Use Poisson when events can occur any number of times in a continuous interval with no fixed upper bound — particularly when events are rare or when n is very large and p is small. Rule of thumb: if n ≥ 20 and p ≤ 0.05 (or np ≤ 10), Poisson is a good approximation to binomial.
What is the Poisson approximation to the binomial?
When n is large (≥ 20) and p is small (≤ 0.05), Binomial(n, p) ≈ Poisson(λ = np). This approximation was historically important before computers, since computing C(n,k) for large n was impractical. Today it remains useful conceptually and when p is so small that the binomial probabilities are difficult to estimate from data. Use the Approximation tab above to compare both distributions numerically.
Why are the mean and variance both equal to λ?
This follows directly from the moment-generating function M(t) = exp(λ(et − 1)). Taking the first derivative at t = 0 gives E[X] = λ. The second cumulant (variance) is also λ. This equal mean-variance property is useful in practice: you can quickly test whether count data follows a Poisson distribution by checking whether the sample mean and sample variance are approximately equal. If variance ≫ mean, consider a negative binomial model.
What is the relationship between Poisson and exponential distributions?
If events occur according to a Poisson process with rate λ, then the waiting time between successive events is Exponential(λ) with mean 1/λ. For instance, if a server receives λ = 10 requests per second (Poisson), the average gap between requests is 0.1 seconds, and that gap is exponentially distributed. This connection underpins M/M/1 queuing theory — M stands for "Memoryless," referring to the exponential inter-arrival and service times.
What are real-world applications of the Poisson distribution?
Poisson distribution applications include: queuing theory and call center staffing (customer arrivals per hour), network traffic engineering (packet arrivals per millisecond), epidemiology (cases per 100,000 population per year), nuclear physics (radioactive decay counts per second), insurance actuarial science (claims per policy per year), website load testing (requests per second), manufacturing quality control (defects per unit area), ecology (organism counts per sampling plot), and astronomy (photons per detector pixel per exposure).

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