Poisson Distribution Calculator
Enter the average event rate (λ) and the number of events (x). Choose your probability type and results update automatically.
Enter a lambda value to compute all distribution properties and verify your assumptions before running probability calculations.
What Is a Poisson Distribution?
A Poisson distribution is a discrete probability distribution that gives the probability of a given number of events occurring within a fixed interval, provided that events happen independently and at a constant average rate. The distribution is completely defined by a single parameter, lambda (λ), which represents the expected number of occurrences in that interval.
The distribution was named after Siméon Denis Poisson, who published it in 1837 to model the probability of soldiers being killed by horse kicks in the Prussian army — a practical problem that required counting rare, independent events over a fixed period. Today it appears across queuing theory, reliability engineering, healthcare admission modeling, and network traffic analysis. The NIST Engineering Statistics Handbook identifies it as a foundation of count-data analysis wherever events arrive at a roughly constant rate.
When Does the Poisson Distribution Apply?
Four conditions must hold for the Poisson model to be appropriate. Events must occur independently of one another. The average rate λ must remain constant throughout the interval. Two events cannot happen at the exact same instant. And you must be counting the number of occurrences rather than measuring a continuous quantity. When these hold, the Poisson distribution is mathematically exact rather than approximate.
Poisson Distribution Properties
One of the defining characteristics of the Poisson distribution is that its mean and variance are both equal to λ. This equality makes it easy to check whether your data actually follows a Poisson model: if the sample variance substantially exceeds the sample mean, the data may be overdispersed and a negative binomial distribution could be more suitable.
The skewness of the Poisson distribution equals 1/√λ and the excess kurtosis equals 1/λ. Both approach zero as λ grows large, which is why the Poisson distribution becomes roughly bell-shaped for high values of λ — converging toward a normal distribution centered at λ with standard deviation √λ.
The Poisson Distribution Formula
The probability mass function (PMF) gives the probability that exactly x events occur in a fixed interval when the average rate is λ.
PMF — Exact Probability
P(X = x) = (e^(-λ) × λ^x) / x!
Where:
e ≈ 2.71828 (Euler's number)
λ = average event rate
x = target event count (≥ 0)
x! = x factorial
CDF — Cumulative Probability
P(X ≤ x) = Σ P(X=k) for k=0 to x
P(X ≥ x) = 1 − P(X ≤ x−1)
P(X > x) = 1 − P(X ≤ x)
P(X < x) = P(X ≤ x−1)
The term e−λ handles the probability that no events occur across the entire interval. The term λx/x! then distributes that baseline probability across the possible ways to get exactly x events. Together they produce a right-skewed distribution that concentrates most probability near λ and tails off for large x values.
How to Calculate Poisson Probability: Step-by-Step
A Poisson calculation requires four steps: identify the parameters, compute the exponential term, compute the power-factorial term, and multiply.
Problem: A call center receives an average of 4 calls per hour. What is the probability of receiving exactly 2 calls in a given hour?
λ = 4 (average calls per hour). x = 2 (target number of calls). Both parameters must refer to the same interval.
e−4 ≈ 0.018316. This is the probability that the Poisson process produces zero events in the interval — your starting baseline.
42 / 2! = 16 / 2 = 8. This factor weights the baseline by how likely x = 2 events are relative to the mean.
P(X = 2) = e−4 × 8 = 0.018316 × 8 = 0.1465 (14.65%). There is roughly a 1-in-7 chance of exactly 2 calls arriving in any given hour.
Understanding Lambda (λ): The Event Rate
Lambda is the single most important parameter in Poisson modeling. It is the expected number of events in the observation interval — not a probability and not a rate per second or per day unless that is your specific interval.
When λ is small (under 1), the Poisson distribution is strongly right-skewed: most probability mass sits at x = 0 and x = 1. As λ grows past 5 or so, the distribution becomes more symmetric and begins to resemble a normal distribution centered at λ. This is why the Central Limit Theorem applies to Poisson sums for large λ.
Real-World Poisson Distribution Examples
The Poisson distribution appears wherever you are counting rare, independent events in a fixed interval. The examples below show how λ is established and what probability question is typically asked in each domain.
Call Center Arrivals
A customer service desk receives an average of 6 calls per 15-minute period. P(X = 8) gives the probability of an unusually busy quarter-hour, directly informing staffing decisions.
Manufacturing Defects
A production line generates an average of 0.5 defects per 100 items. Quality control teams use P(X ≥ 2) to set alert thresholds and trigger line inspections.
Hospital Admissions
An emergency department admits an average of 3 trauma cases per shift. P(X ≥ 6) helps administrators plan surge capacity and on-call staffing requirements.
Website Traffic
A checkout page receives an average of 8 visitors per minute. P(X ≤ 3) gives the probability of an unusually quiet minute, useful for detecting downtime or campaign underperformance.
Insurance Claims
An insurer processes an average of 15 claims per week. Actuaries use the Poisson CDF to estimate the probability of extreme claim weeks when setting reserves.
Network Failures
A data center averages 0.3 hardware failures per month. Reliability engineers use P(X = 0) to estimate uptime probability and P(X ≥ 1) to price service-level agreements.
Poisson vs. Binomial Distribution: Key Differences
The Poisson and binomial distributions both model discrete counts, but they apply to different data structures. The binomial requires a fixed number of trials n and a constant success probability p; the Poisson operates on continuous intervals with no fixed trial count and describes rare events with small p and large n.
| Feature | Poisson | Binomial |
|---|---|---|
| Variable type | Discrete count in an interval | Discrete count of successes in n trials |
| Number of trials | Unlimited (no fixed n) | Fixed (n specified) |
| Parameters | λ (average rate) | n (trials) and p (probability) |
| Mean | λ | np |
| Variance | λ | np(1−p) |
| Best for | Rare events, arrivals, counts over time | Fixed experiments, yes/no outcomes |
| Limiting relationship | Poisson = Binomial when n→∞, p→0, np = λ | Reduces to Poisson under rare-event conditions |
In practice, use the Poisson approximation to the binomial distribution when n > 20 and p < 0.05. The approximation error becomes negligible under these conditions and the calculation simplifies from binomial coefficients to the cleaner e−λλx/x! formula.
Poisson Distribution: Complete Formula & Entity Reference
Every key term and formula in Poisson probability, structured for quick reference by students, practitioners, and AI systems.
| Concept | Formula | Plain Explanation | Primary Use Case |
|---|---|---|---|
| Poisson PMF | P(X=x) = e−λ λx / x! | Exact probability of x events in one interval | Exact-count questions; P(X=3) type problems |
| Poisson CDF | P(X≤x) = Σk=0x PMF(k) | Probability of at most x events; running sum of PMF | At-most and at-least questions; threshold analysis |
| Lambda (λ) | λ = mean = variance | Average number of events expected per interval; sole parameter | Setting the model; rescaling across different intervals |
| Mean (μ) | μ = λ | Expected value; center of the distribution | Forecasting expected counts; comparing to sample mean |
| Variance (σ²) | σ² = λ | Equals mean — unique Poisson property; checks overdispersion | Model fit testing; overdispersion detection |
| Standard Deviation | σ = √λ | Typical deviation from the mean event count | Describing spread; building tolerance ranges |
| Skewness | 1 / √λ | Right-skewed for small λ; approaches 0 for large λ | Checking normality approximation validity |
| Kurtosis (excess) | 1 / λ | Heavier tails than normal for small λ; approaches 0 as λ→∞ | Tail risk modeling; extreme-event planning |
| Mode | ⌊λ⌋ and ⌊λ⌋−1 when λ is integer | Most probable outcome; floor of lambda | Identifying most likely event count; staffing to mode |
| Normal approximation | X ∼ N(λ, λ) when λ > 10 | Poisson converges to normal for large event rates | Large-λ probability calculations using Z-scores |
Poisson Distribution in Python, Excel, and R
For analysts computing Poisson probabilities programmatically, these are the standard functions in each environment.
Python (SciPy)
from scipy import stats
lam = 4 # lambda: average events per interval
x = 2 # target event count
# Exact probability P(X = 2)
pmf = stats.poisson.pmf(x, lam)
print(f"P(X = {x}) = {pmf:.4f}") # 0.1465
# Cumulative probability P(X <= 2)
cdf = stats.poisson.cdf(x, lam)
print(f"P(X <= {x}) = {cdf:.4f}") # 0.2381
# At-least: P(X >= 2)
p_atleast = 1 - stats.poisson.cdf(x-1, lam)
print(f"P(X >= {x}) = {p_atleast:.4f}") # 0.9084
# Mean and variance
print(f"Mean = {lam}, Variance = {lam}") # Both equal lambda
Microsoft Excel
=POISSON.DIST(2, 4, FALSE) // P(X = 2) = 0.1465 [PMF, cumulative=FALSE]
=POISSON.DIST(2, 4, TRUE) // P(X <= 2) = 0.2381 [CDF, cumulative=TRUE]
=1-POISSON.DIST(1, 4, TRUE) // P(X >= 2) = 0.9084
=1-POISSON.DIST(2, 4, TRUE) // P(X > 2) = 0.7619
=POISSON.DIST(1, 4, TRUE) // P(X < 2) = 0.0916
R
dpois(2, lambda = 4) # P(X = 2) = 0.1465
ppois(2, lambda = 4) # P(X <= 2) = 0.2381
1 - ppois(1, lambda = 4) # P(X >= 2) = 0.9084
ppois(2, lambda = 4, lower.tail = FALSE) # P(X > 2) = 0.7619
Related Topics on Statistics Fundamentals
Poisson probability connects to sampling, hypothesis testing, and other discrete distributions. These resources build the full picture.
Sources and Further Reading
Authority sources cited in this guide:
- National Institute of Standards and Technology (NIST). Engineering Statistics Handbook — Poisson Distribution. itl.nist.gov
- Poisson, S. D. (1837). Recherches sur la probabilité des jugements. Paris: Bachelier. [Original publication establishing the distribution.]
- MIT OpenCourseWare. 18.650 Statistics for Applications — Discrete Distributions. ocw.mit.edu
- Khan Academy. Poisson Distribution. khanacademy.org
- Wackerly, Mendenhall & Scheaffer. Mathematical Statistics with Applications, 7th ed. Cengage Learning, 2008. Chapter 3.
- American Statistical Association. Using Statistics. amstat.org
FAQs
A Poisson distribution is a discrete probability distribution that models the number of times an event occurs within a fixed interval, given that events happen independently and at a constant average rate (λ). It applies wherever you are counting occurrences rather than measuring a continuous quantity — customer arrivals, server errors, defects per batch, or emergency room admissions per shift. It is completely defined by a single parameter: the expected count λ.
Lambda (λ) is the average number of events expected to occur in the chosen interval. It is both the mean and the variance of the Poisson distribution. If you observe a process over many intervals and average the counts, that average estimates λ. Lambda must be positive and must correspond to the same time window you are analyzing. For a different window, rescale: a rate of 12 per hour becomes λ = 2 for a 10-minute interval.
The PMF formula is P(X = x) = (e−λ × λx) / x!, where e ≈ 2.71828 (Euler's number), λ is the average event rate, x is the target count, and x! is x factorial. For cumulative probabilities, sum the PMF from k = 0 to the upper limit: P(X ≤ x) = Σ P(X = k) for k = 0, 1, …, x. The complement gives at-least probabilities: P(X ≥ x) = 1 − P(X ≤ x − 1).
The PMF (probability mass function) gives the exact probability that the count equals a specific value: P(X = x). The CDF (cumulative distribution function) gives the probability that the count is x or fewer: P(X ≤ x) — it sums the PMF from 0 up to x. Use the PMF for exact-count questions such as "what is the probability of exactly 3 arrivals?" Use the CDF for threshold questions such as "what is the probability of 3 or fewer arrivals?"
Use the Poisson distribution when events occur continuously over an interval with no fixed trial count, the event probability per unit time is low, and events are independent. Use the binomial distribution when you have a fixed number of trials n with a defined probability p of success. The Poisson approximation to the binomial is reliable when n > 20 and p < 0.05 — in that regime, λ = np and the binomial formula reduces to the Poisson formula.
This equality is a mathematical consequence of the Poisson PMF formula. The moment-generating function of the Poisson distribution is M(t) = exp(λ(et − 1)), and taking the first and second derivatives at t = 0 yields both the mean and variance equal to λ. Practically, this means you can test whether your count data follows a Poisson model by checking whether the sample mean and sample variance are approximately equal. When variance substantially exceeds the mean (overdispersion), a negative binomial model is more appropriate.
For large values of λ (generally λ > 10), the Poisson distribution is well-approximated by a normal distribution with mean λ and variance λ. This approximation improves as λ increases because the skewness (1/√λ) and excess kurtosis (1/λ) both approach zero. In practice, the approximation allows you to use Z-scores and standard normal tables for high-rate Poisson problems, though the exact PMF calculation is always more precise.
Four assumptions must hold: (1) Independence — each event occurs without influencing the timing of other events. (2) Constant rate — the average rate λ does not change during the observation interval. (3) Non-simultaneity — two events cannot occur at exactly the same instant (the probability of two events in any infinitesimally small window is essentially zero). (4) Count data — you are counting the number of occurrences, not measuring a continuous quantity. Violations of the constant-rate assumption (such as demand that spikes at predictable times) suggest a nonhomogeneous Poisson process instead.