What Is the Poisson Distribution?
The distribution was named after French mathematician Siméon Denis Poisson, who described it in his 1837 work Recherches sur la probabilité des jugements. His interest was in modeling the number of wrongful convictions in a judicial system — a rare, independent event occurring at some background rate, which is exactly the kind of phenomenon the distribution handles well.
Practically, the Poisson distribution fits any process where you're counting occurrences of an event over a continuous medium (time, length, area) and the events don't bunch up or repel each other. Customer arrivals, network packets, equipment failures, radioactive decay events, and typing errors all fit this pattern. The single parameter that drives everything is λ — the expected number of events in your chosen interval.
The Poisson distribution is a discrete probability distribution that models the number of times an independent event occurs in a fixed interval when the average rate of occurrence λ is known and constant. Its probability mass function is P(X = k) = λᵏe⁻λ / k!, where k = 0, 1, 2, … and e ≈ 2.71828. Both the mean and variance equal λ. Common applications include queueing theory, network traffic modeling, epidemiology, and manufacturing quality control.
The Poisson Distribution Formula (PMF)
The probability mass function (PMF) of the Poisson distribution gives the probability that a Poisson random variable X takes exactly the value k:
P(X = k) = probability of exactly k events
λ = average event rate (lambda)
e ≈ 2.71828 (Euler's number)
k = number of events (0, 1, 2, …)
k! = k factorial (k × (k−1) × … × 1)
Breaking Down Each Component
λ (lambda): The rate parameter. It is the expected — meaning average — number of events in your interval. If a call center receives 300 calls in 50 minutes, then λ = 6 calls per minute. λ must be strictly positive.
k: The specific count you're asking about. k must be a non-negative integer: 0, 1, 2, 3, and so on. There is no upper bound on k, though probabilities become vanishingly small as k grows far above λ.
e⁻λ: This term is the probability of zero events occurring — it's the "baseline" from which all other probabilities are built. For λ = 3, e⁻³ ≈ 0.0498, meaning roughly a 5% chance of observing no events at all.
λᵏ: Captures how the probability shifts as we consider larger or smaller counts relative to the rate.
k! (k factorial): Accounts for the number of ways k events could have occurred, preventing double-counting of equivalent outcomes.
Cumulative Distribution Function (CDF)
The CDF gives the probability of at most k events, summing over all values from 0 through k:
P(X ≤ k) = probability of k or fewer events
Σ = sum from i = 0 to k
For example, to find P(X ≤ 2) when λ = 3, you sum P(X = 0) + P(X = 1) + P(X = 2). The CDF is what you use in practice when you want "at most" or "at least" questions: P(X ≥ k) = 1 − P(X ≤ k − 1).
Poisson PMF: Probability for Each Count (λ = 4)
The 4 Assumptions of the Poisson Distribution
The Poisson model produces accurate probabilities only when all four conditions hold. Checking them before applying the distribution is not optional — skipping this step is one of the most common errors in applied statistics.
Independence
Events must occur independently. The occurrence of one event must not change the probability of the next. A car accident at one intersection does not cause another at a different intersection — those counts are independent. But a server crash that triggers a flood of support tickets violates this assumption because one event directly causes many others.
Constant Rate (λ is Fixed)
The average event rate λ must remain constant across the entire interval. Customer arrivals at a grocery store are not Poisson-distributed across an entire day because the rate spikes at lunch and drops at 3 AM. You can still use Poisson — but you need to define your interval carefully so that λ stays stable within it (for example, "arrivals between 11 AM and noon").
No Simultaneous Events
Two events cannot occur at the exact same instant. This is almost always satisfied in practice: even when two customers seem to arrive simultaneously, there is still some infinitesimally small time gap. The formal statement is that the probability of two or more events in an infinitesimally small interval is negligible — approaching zero faster than the length of the interval.
Proportionality
The probability of an event in a small sub-interval is proportional to the length of that sub-interval. If λ = 6 events per hour, then in any 10-minute slice the expected count is 1 event. This means the Poisson process is memoryless: the waiting time since the last event tells you nothing about when the next one will arrive.
If your observed data shows variance substantially larger than the mean, the Poisson model may not fit — this is called overdispersion. Common causes: events cluster (contagion), the rate λ varies across subgroups, or there are excess zeros. A negative binomial distribution is often a better choice in these situations.
Key Properties of the Poisson Distribution
- Mean (Expected Value): E(X) = λ
- Variance: Var(X) = λ
- Standard Deviation: σ = √λ
- Skewness: 1/√λ — always positive (right-skewed), approaches symmetry as λ increases
- Kurtosis (excess): 1/λ — more peaked than normal for small λ
- Support: k = 0, 1, 2, 3, … (unbounded above)
- Additive property: If X ~ Poisson(λ₁) and Y ~ Poisson(λ₂) are independent, then X + Y ~ Poisson(λ₁ + λ₂)
- Relationship to exponential: If inter-event times are Exponential(λ), then event counts per interval are Poisson(λ)
Why Mean = Variance = λ
This property is distinctive — no other common distribution shares it. It follows from the moment generating function (MGF) of the Poisson distribution: M(t) = e^(λ(eᵗ − 1)). Taking the first derivative at t = 0 gives E(X) = λ; the second central moment gives Var(X) = λ.
In practice, this equality is a diagnostic tool. If you collect count data and compute the sample mean and sample variance, and they are close to each other, a Poisson model is a reasonable candidate. If the variance is much larger than the mean, the data shows overdispersion. If the variance is much smaller than the mean (underdispersion), events may be self-correcting — a property better captured by the binomial distribution with a moderate p.
The Poisson Process
The Poisson distribution arises naturally from a Poisson process, which is a stochastic model for counting random, independent events over continuous time. If events arrive according to a Poisson process with rate λ, then the number of arrivals in any interval of length t follows a Poisson distribution with parameter λt. This scaling property makes the model flexible: knowing the hourly rate λ = 6 lets you immediately compute the distribution for any sub-interval — in 10 minutes, the count follows Poisson(1).
The Poisson process is closely related to the binomial distribution. Formally, the Poisson distribution is the limiting case of Binomial(n, p) as n → ∞ and p → 0, with the product np = λ held constant. This connection is why the Poisson distribution is often used to approximate binomial probabilities when n is large (>100) and p is small (<0.01).
Poisson Distribution Calculator
Enter λ (average rate) and k (number of events) to compute P(X = k), the cumulative probability P(X ≤ k), and the complement P(X > k).
Poisson Probability Calculator
8 Fully Worked Poisson Distribution Examples
Each example identifies λ, specifies k, applies the PMF formula step by step, and states a practical conclusion. Examples span healthcare, manufacturing, telecommunications, traffic, ecology, and web analytics.
Example 1 — Hospital Emergency Arrivals
A hospital emergency room records an average of 4 patient arrivals per hour. What is the probability that exactly 2 patients arrive in a given hour?
Identify parameters: λ = 4 (average arrivals per hour), k = 2 (the count we want). The 4 assumptions hold: arrivals are independent, the rate is stable for this hour, two patients cannot arrive at the exact same instant.
Calculate e⁻λ: e⁻⁴ = 1/e⁴ = 1/54.598 ≈ 0.01832
Calculate λᵏ: 4² = 16
Calculate k!: 2! = 2 × 1 = 2
Apply the formula:
P(X = 2) = (16 × 0.01832) / 2 = 0.29312 / 2 = 0.1465
✅ There is a 14.65% probability of exactly 2 patient arrivals in a given hour when the average rate is 4 per hour.
Example 2 — Manufacturing Defects
A factory produces machine components with an average of 2 defects per 500-unit batch. What is the probability that a batch contains no defects?
Parameters: λ = 2 (average defects per batch), k = 0 (zero defects). The defect rate is low and defects are independent — Poisson applies.
Apply formula for k = 0: Note that λ⁰ = 1 and 0! = 1, so P(X = 0) = e⁻λ.
Calculate:
P(X = 0) = e⁻² = 1/e² ≈ 1/7.389 ≈ 0.1353
✅ About 13.53% of batches will be defect-free. The probability of zero defects is simply e⁻λ for any Poisson random variable.
Example 3 — Call Center Arrivals (At Least k Events)
A call center receives an average of 6 calls per minute. What is the probability of receiving at least 8 calls in a given minute?
Parameters: λ = 6, and we want P(X ≥ 8) = 1 − P(X ≤ 7). We must compute the CDF up to k = 7 and subtract from 1.
Compute P(X = k) for k = 0 to 7, λ = 6:
e⁻⁶ ≈ 0.002479
P(0) = 0.002479 | P(1) = 0.01488 | P(2) = 0.04463 | P(3) = 0.08924
P(4) = 0.1339 | P(5) = 0.1606 | P(6) = 0.1606 | P(7) = 0.1377
Sum for CDF: P(X ≤ 7) = 0.002479 + 0.01488 + 0.04463 + 0.08924 + 0.1339 + 0.1606 + 0.1606 + 0.1377 = 0.7440
Complement: P(X ≥ 8) = 1 − 0.7440 = 0.2560
✅ There is a 25.6% probability of receiving 8 or more calls in a given minute. The call center should staff for this scenario roughly one in every four minutes.
Example 4 — Road Traffic Accidents
A particular road junction averages 1.5 accidents per week. What is the probability of exactly 3 accidents in a given week?
Parameters: λ = 1.5, k = 3. Accidents occur independently and randomly, so the Poisson model is appropriate.
Compute:
λᵏ = 1.5³ = 3.375
e⁻λ = e⁻¹·⁵ ≈ 0.22313
k! = 3! = 6
Apply formula:
P(X = 3) = (3.375 × 0.22313) / 6 = 0.75307 / 6 ≈ 0.1255
✅ There is a 12.55% probability of exactly 3 accidents in a week at this junction. Traffic engineers use this to assess whether safety interventions have changed the accident rate.
Example 5 — Web Server Requests
A web server receives an average of 300 requests per second. What is the probability of receiving exactly 310 requests in a given second? (Use the normal approximation.)
Parameters: λ = 300. Since λ is large (>30), the Poisson distribution is approximately normal: X ~ N(μ = 300, σ² = 300), so σ = √300 ≈ 17.32.
Apply continuity correction: For P(X = 310) using the normal approximation, use P(309.5 < Y < 310.5).
Standardize:
z₁ = (309.5 − 300) / 17.32 = 0.549
z₂ = (310.5 − 300) / 17.32 = 0.606
Look up z-values (see the z-table):
P(Z < 0.606) − P(Z < 0.549) ≈ 0.7277 − 0.7083 = 0.0194
✅ About 1.94% — there is roughly a 2% chance of exactly 310 requests in a second. For large λ, software tools give more precise answers than the PMF calculation, which requires computing 300!.
Example 6 — Email Spam Arrivals
A spam filter logs an average of 12 spam emails per hour. What is the probability of receiving fewer than 10 spam emails in a given hour?
Parameters: λ = 12, we want P(X < 10) = P(X ≤ 9).
e⁻¹² ≈ 0.000006144
Sum P(X = k) for k = 0 to 9 using the PMF formula for each k, with λ = 12.
P(0) ≈ 0.0000061 | P(1) ≈ 0.0000738 | P(2) ≈ 0.000443 | P(3) ≈ 0.001773
P(4) ≈ 0.005318 | P(5) ≈ 0.01276 | P(6) ≈ 0.02553 | P(7) ≈ 0.04376
P(8) ≈ 0.06564 | P(9) ≈ 0.08752
Sum: P(X ≤ 9) ≈ 0.0000061 + 0.0000738 + 0.000443 + 0.001773 + 0.005318 + 0.01276 + 0.02553 + 0.04376 + 0.06564 + 0.08752 = 0.2424
✅ There is roughly a 24.2% probability that fewer than 10 spam emails arrive in a given hour. The system should still expect 12 on average — the below-average hours are more common than intuition suggests.
Example 7 — Rescaling λ to a Different Interval
A bank ATM has an average of 18 transactions per hour. What is the probability of exactly 5 transactions in a 15-minute window?
Rescale λ: The rate is 18 per hour. For a 15-minute window (¼ hour): λ = 18 × (15/60) = 18 × 0.25 = 4.5
Parameters: λ = 4.5, k = 5
Apply formula:
e⁻⁴·⁵ ≈ 0.011109
λᵏ = 4.5⁵ = 1845.56
k! = 5! = 120
P(X = 5) = (1845.56 × 0.011109) / 120 = 20.50 / 120 ≈ 0.1708
✅ There is a 17.08% probability of exactly 5 ATM transactions in any 15-minute window. This rate-scaling property — multiplying λ by the fraction of the original interval — works for any Poisson process.
Example 8 — Ecology: Plants Per Quadrat
An ecologist counting rare orchid plants finds an average of 0.8 plants per 1-square-meter quadrat. What is the probability of finding exactly 1 plant in a randomly chosen quadrat?
Parameters: λ = 0.8, k = 1. The Poisson distribution applies to spatial counts just as readily as time-based counts — the "interval" is area rather than time.
Apply formula:
e⁻⁰·⁸ ≈ 0.4493
λ¹ = 0.8
k! = 1! = 1
P(X = 1) = (0.8 × 0.4493) / 1 = 0.3595
✅ There is a 35.95% probability of finding exactly 1 orchid plant in a 1 m² quadrat. Note that P(X = 0) = e⁻⁰·⁸ ≈ 0.449, so finding no plants is actually the most likely single outcome — because the average is less than 1.
Real-World Applications of the Poisson Distribution
The Poisson distribution is one of the most applied distributions in quantitative fields. Below are eight industries where it appears routinely, with the specific measurement being modeled and a typical λ value.
Healthcare
Emergency department arrivals, ICU admissions, drug side-effect reports, disease incidence rates per 100,000 population. Epidemiologists use Poisson regression to model disease counts.
Manufacturing
Defect counts per unit, machine breakdowns per shift, contamination events per production run. Poisson control charts (c-charts) monitor defect rates in quality control.
Telecommunications
Network packet arrivals, call center demand, dropped calls per hour, SMS delivery failures. Poisson arrival models are the foundation of queueing theory (Erlang-B and Erlang-C formulas).
Traffic Engineering
Vehicle arrivals at intersections, accident counts per road segment per year, pedestrian crossings per hour. Traffic signal timing is often derived from Poisson arrival assumptions.
Data Science & ML
Modeling rare events, click-through rates for low-frequency ads, user actions per session, error log counts. Poisson regression is a generalized linear model for count outcomes.
Insurance & Finance
Insurance claims per policy per year, loan default counts, stock market extreme events. Actuaries use compound Poisson processes to model aggregate claims in non-life insurance.
Physics
Radioactive decay events per second (the original application by Rutherford and Geiger, 1910), photon counts in astronomy, cosmic ray detections per detector per hour.
Ecology
Species counts per sampling unit, plant density per quadrat, animal sightings per survey transect. Poisson models test whether species are randomly distributed or clustered/overdispersed.
Poisson vs Other Distributions
Poisson vs Binomial Distribution
| Feature | Poisson | Binomial |
|---|---|---|
| Data type | Count in a continuous interval | Count of successes in n trials |
| Parameters | λ (rate) | n (trials), p (success probability) |
| Fixed trials? | No — unbounded count | Yes — exactly n trials |
| Mean | λ | np |
| Variance | λ | np(1 − p) |
| Mean = Variance? | Always | Only if p = 0 |
| When to use | Rare events, large population, no fixed n | Fixed n, constant p, two outcomes |
| Relationship | Limiting case of Binomial as n → ∞, p → 0, np = λ | Becomes Poisson at the limit |
A practical rule of thumb: approximate the binomial with a Poisson distribution when n > 100 and p < 0.01. For example, if 0.5% of 200 widgets are defective (n = 200, p = 0.005), then λ = np = 1 gives a Poisson approximation that is accurate to three decimal places. See the full binomial distribution guide for more detail on when to switch models.
Poisson vs Normal Distribution
| Feature | Poisson | Normal |
|---|---|---|
| Data type | Discrete (counts: 0, 1, 2, …) | Continuous (any real number) |
| Support | Non-negative integers only | All real numbers (−∞ to +∞) |
| Parameters | λ | μ (mean), σ² (variance) |
| Symmetry | Right-skewed for small λ | Perfectly symmetric |
| Approximation | Approximated by Normal when λ > 30 | Can approximate Poisson for large λ |
| Negative values? | Impossible | Possible (can be an issue for count models) |
When λ ≥ 30, the Poisson distribution is approximately bell-shaped, and the normal approximation P(X ≤ k) ≈ Φ((k + 0.5 − λ)/√λ) is accurate. The continuity correction (adding 0.5) improves the approximation at the boundaries. For smaller λ, the distribution is clearly right-skewed and the normal approximation fails. See the normal distribution guide for the full approximation procedure.
Poisson vs Exponential Distribution
| Feature | Poisson | Exponential |
|---|---|---|
| What it models | Number of events in an interval | Time between consecutive events |
| Data type | Discrete | Continuous |
| Parameter | λ (events per unit time) | λ (same rate — events per unit time) |
| Mean | λ (events per interval) | 1/λ (mean waiting time) |
| Connection | If counts are Poisson(λ), … | …then inter-arrival times are Exp(λ) |
The Poisson and exponential distributions are two sides of the same coin. If events arrive according to a Poisson process, the count in any fixed interval follows Poisson(λ), while the time you wait for the next event follows Exponential(λ). This duality is used extensively in queueing theory and reliability engineering.
When to Use the Poisson Distribution
Distribution Selection Decision Tree
Poisson Distribution in R, Python, and Excel
R
R provides built-in functions for the Poisson PMF (dpois), CDF (ppois), quantile function (qpois), and random sampling (rpois):
dpois(x = 2, lambda = 4) # returns 0.1465
# CDF: P(X ≤ 5) for lambda = 4
ppois(q = 5, lambda = 4) # returns 0.7851
# P(X >= 8) for lambda = 6
1 - ppois(q = 7, lambda = 6) # returns 0.2560
# Generate 100 random Poisson values with lambda = 3
rpois(n = 100, lambda = 3)
Python (SciPy)
# PMF: P(X = 2) when lambda = 4
poisson.pmf(k=2, mu=4) # 0.14652
# CDF: P(X <= 5) when lambda = 4
poisson.cdf(k=5, mu=4) # 0.78513
# Survival: P(X > 7) when lambda = 6
poisson.sf(k=7, mu=6) # 0.25602
# Random sample of 100 values
poisson.rvs(mu=3, size=100)
Excel
| Goal | Excel Formula | Example (λ = 4, k = 2) |
|---|---|---|
| P(X = k) — PMF | =POISSON.DIST(k, lambda, FALSE) | =POISSON.DIST(2, 4, FALSE) → 0.1465 |
| P(X ≤ k) — CDF | =POISSON.DIST(k, lambda, TRUE) | =POISSON.DIST(5, 4, TRUE) → 0.7851 |
| P(X > k) | =1 - POISSON.DIST(k, lambda, TRUE) | =1 - POISSON.DIST(7, 6, TRUE) → 0.2560 |
The COUNT Framework for Applying Poisson
COUNT: A 5-Step Poisson Application Guide
Check all four: independence, constant rate, no simultaneous events, and proportionality. If any assumption is violated, the model may not apply.
Identify or estimate λ from historical data, published rates, or a pilot study. Remember to define the interval clearly — λ must match the interval you're analyzing.
Ask whether one event influences the next. If events cluster or repel each other, reconsider the model. Check that variance ≈ mean in your sample data.
Decide on k — the specific count (or range) you want a probability for. Are you asking for exactly k, at most k, or at least k events?
Apply P(X = k) = λᵏe⁻λ / k! (or the CDF for cumulative questions). Translate the numerical result into a plain-language conclusion for your audience.
Common Mistakes with the Poisson Distribution
| Mistake | What Goes Wrong | Correct Approach |
|---|---|---|
| Using Poisson when events are not independent | Underestimates probabilities for high-count events | Verify independence; if events cluster, use negative binomial |
| Ignoring the interval definition | λ from one interval applied to a different interval without scaling | Rescale: if λ = 12/hr and interval is 30 min, use λ = 6 |
| Computing P(X > k) as 1 − P(X = k) | Severely wrong — omits the probability of all values below k too | P(X > k) = 1 − P(X ≤ k) — use the CDF, not just the PMF |
| Applying Poisson when n and p are both moderate | Approximation error; binomial gives the exact answer | Use Poisson only when n > 100 and p < 0.01 for the approximation |
| Treating the Poisson mean as the most probable value | For small λ, the mode is below the mean (mode = ⌊λ⌋ if λ is not an integer) | Check the PMF table — find the k with the highest P(X = k) |
Poisson Distribution — Entity & Formula Glossary
| Term / Symbol | Definition |
|---|---|
| Poisson Distribution | Discrete probability distribution for counts of independent events in a fixed interval, parameterized by λ |
| λ (Lambda) | Average number of events in the interval; the sole parameter; equals both mean and variance |
| P(X = k) | Probability mass function: λᵏe⁻λ / k! |
| P(X ≤ k) | Cumulative distribution function: sum of P(X = i) for i = 0 to k |
| e | Euler's number ≈ 2.71828; base of the natural logarithm |
| k! | k factorial: product of all positive integers up to k (0! = 1 by convention) |
| Mean E(X) | λ — the expected number of events |
| Variance Var(X) | λ — equals the mean; diagnostic for checking model fit |
| Standard Deviation σ | √λ |
| Overdispersion | When observed variance > mean; indicates Poisson may not fit |
| Poisson Process | Continuous-time stochastic process where event counts in any interval follow a Poisson distribution |
| Poisson Regression | Generalized linear model for count outcome variables using a log link function |
| Siméon Denis Poisson | French mathematician (1781–1840) who derived the distribution in 1837 |
Frequently Asked Questions
People Also Ask
What is the Poisson distribution used for?
The Poisson distribution models the number of times a random, independent event occurs in a fixed interval when you know its average rate. Typical uses: counting emergency room arrivals per hour, manufacturing defects per batch, calls to a hotline per minute, traffic accidents per month, or email spam messages per day. Any process generating count data over a continuous medium (time, area, volume) where events are independent is a candidate for the Poisson model.
People Also Ask
How is the Poisson distribution related to the binomial distribution?
The Poisson distribution is the mathematical limit of the binomial distribution when the number of trials n approaches infinity and the success probability p approaches zero, while the product np = λ remains constant. This means you can use a Poisson distribution to approximate binomial probabilities whenever n is large (>100) and p is small (<0.01). The approximation is accurate to several decimal places in those conditions.
People Also Ask
Can λ be a non-integer (decimal) value?
Yes — λ can be any positive real number, including decimals. The average number of events in an interval rarely works out to a whole number. A hospital might average 3.7 admissions per hour, or a factory might average 0.4 defects per batch. The PMF formula handles any positive real λ exactly. The integer k in the formula is the specific count you're asking about — not λ.
People Also Ask
What is the mode of the Poisson distribution?
If λ is not an integer, the mode (most probable value) is ⌊λ⌋ — the largest integer less than or equal to λ. If λ is itself an integer, there are two modes: λ and λ − 1, both with equal probability. For example, with λ = 4: modes are 4 and 3, both giving P(X = k) = 0.1954. With λ = 4.5: mode is 4 only, giving P(X = 4) = 0.1898.
People Also Ask
How do you know if data follows a Poisson distribution?
Three practical checks: (1) Compare the sample mean and sample variance — they should be approximately equal. A large ratio (variance/mean, called the dispersion index) above 1 suggests overdispersion. (2) Plot the observed frequency distribution against the theoretical Poisson PMF for the estimated λ. (3) Run a goodness-of-fit test — either a chi-square test or a Kolmogorov-Smirnov test — to formally assess fit. See normality and distribution tests for the general approach.