Poisson Distribution Calculator
Work out how likely it is to observe exactly, at most, at least or between given numbers of events in a fixed window of time or space, such as arrivals, calls or defects, when you only know the long-run average rate. Includes the mean, variance, a bar chart and a table.
Not sure this is the right model? See binomial vs Poisson vs hypergeometric: three questions that decide.
The average number of events in one window; it need not be a whole number
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Learn More
Poisson Distribution Explained
From an average rate to exact count probabilities: worked call-center example, the assumptions checklist, and the binomial approximation shown numerically.
Probability Distributions
Compare the binomial, Poisson, normal, uniform, and exponential distributions and learn how to choose the right model for counts, measurements, and waiting times.
Before you calculate
- State the rate λ and the count k over the same window: if λ is events per hour, k is a count for one hour.
- λ does not need to be a whole number: an average of 2.7 support tickets per day is a valid rate.
- Check that events occur independently and one at a time; scheduled or clustered arrivals break the model.
When the Poisson model fits
The Poisson distribution describes counts of events that happen randomly at a stable average rate. Three conditions need to hold:
- Independence: one event does not make another more or less likely. A machine breakdown that triggers a cascade of further breakdowns violates this.
- Constant rate: the average number of events per unit of time or space stays the same across the window. Lunchtime rushes mean the hourly rate of a restaurant is not constant across the day.
- No simultaneous events: events arrive one at a time. Group arrivals, such as a bus unloading thirty customers at once, need a batch-arrival model.
Unlike the binomial distribution, there is no fixed number of trials: the count has no hard upper limit, only rapidly shrinking probabilities for large values.
The Poisson probability formula
P(X = k) = λ^k × e^(−λ) / k!
P(X ≤ k) = P(X = 0) + P(X = 1) + … + P(X = k)
Mean = λ Variance = λ Standard deviation = √λ Skewness = 1/√λ
λ is the average number of events per window, k the number of events you ask about, e ≈ 2.71828 and k! the product 1 × 2 × … × k. The equality of mean and variance is the signature of the Poisson distribution. If your observed counts have a variance far larger than their mean (overdispersion), the negative binomial distribution usually fits better. “At least” and “more than” are computed from the upper tail directly, so a rare-event probability such as 10−15 keeps its digits.
Rescaling the rate to your window
The rate scales linearly with the size of the window. If a website averages 12 signups per hour, then over a 15-minute window λ = 12 × 15/60 = 3, and over an 8-hour shift λ = 96. Rescale λ before you enter it rather than rescaling the answer afterward. With λ = 3, the chance of five or more signups in 15 minutes is 0.184737 (choose One value k with k = 5 and read the “at least” card).
The same logic works for space: flaws per square meter of fabric, potholes per mile, typos per page. With an average of 1 typo per page, the chance that a given page is typo-free is P(X = 0) = e−1 ≈ 0.367879, about 37%.
Worked example: custom cake orders
A bakery receives an average of λ = 4 custom cake orders per day. What is the probability of exactly 2 orders tomorrow?
- Write the formula: P(X = 2) = 4² × e−4 / 2!
- Evaluate each piece: 4² = 16, e−4 ≈ 0.018316 and 2! = 2.
- Combine: P(X = 2) = 16 × 0.018316 / 2 ≈ 0.146525, a 14.7% chance.
The cumulative cards answer planning questions. P(X ≤ 2) = 0.018316 + 0.073263 + 0.146525 ≈ 0.238103, so about a 24% chance of a slow day with two orders or fewer. P(X ≥ 2) = 1 − 0.091578 = 0.908422: the bakery can count on at least two orders about 91% of the time. The chance of between 2 and 6 orders inclusive (choose Between two values) is 0.797748. The summary is immediate: mean 4, variance 4 and standard deviation √4 = 2. Load example reproduces all of these numbers.
Excel, R, Python and TI-84
| Software | P(X = k) | P(X ≤ k) |
|---|---|---|
| Excel, Google Sheets | POISSON.DIST(k, λ, FALSE) | POISSON.DIST(k, λ, TRUE) |
| R | dpois(k, λ) | ppois(k, λ) |
| Python (SciPy) | scipy.stats.poisson.pmf(k, λ) | scipy.stats.poisson.cdf(k, λ) |
| TI-84 | poissonpdf(λ, k) | poissoncdf(λ, k) |
For P(X ≥ k) use ppois(k - 1, λ, lower.tail = FALSE) in R or poisson.sf(k - 1, λ) in SciPy. The poissonpdf and poissoncdf calculator works like the TI-84 functions.
Real applications
- Call centers and queues: staffing models start from the distribution of arrivals per interval to predict waiting times.
- Reliability engineering: the number of component failures in a maintenance period drives spare-part inventory.
- Insurance: claim counts per policy year are the classic actuarial application.
- Epidemiology and biology: case counts in a region, mutations per genome segment and radioactive decays per second follow Poisson models when the underlying rate is stable.
Related guides and calculators
The time between Poisson events follows the exponential distribution, and the Poisson distribution approximates the binomial distribution when n is large and p is small. Counts that vary more than a Poisson allows are modeled by the negative binomial distribution. Read the Poisson distribution explained and binomial vs Poisson vs hypergeometric to choose between the models.
Frequently Asked Questions
How is the Poisson distribution different from the binomial?
The binomial counts successes out of a fixed number of trials, so the count is capped at n. The Poisson counts events in a fixed interval with no upper bound, using only the average rate lambda. When trials are numerous and the success probability is small, the binomial converges to a Poisson with lambda = np, which is why the Poisson is often called the law of rare events.
Can lambda be a decimal like 2.5?
Yes. Lambda is an average, not a count, so any positive value is valid. Only k, the number of events you ask about, must be a whole number, because you cannot observe half an event.
What is overdispersion and why does it matter?
A Poisson variable has variance equal to its mean. Real count data often shows a variance well above the mean, for example when events cluster or the rate drifts over time. That mismatch, called overdispersion, makes Poisson probabilities too optimistic in the tails; a negative binomial model is the usual remedy.
How does the Poisson relate to the exponential distribution?
They describe the same process from two angles. The Poisson counts how many events land in a fixed window, while the exponential measures the waiting time between consecutive events. If counts per hour are Poisson with rate lambda, the gaps between events are exponential with mean 1/lambda hours.
When can I approximate a Poisson with a normal distribution?
Once lambda is reasonably large, a common threshold is lambda >= 20, the Poisson is nearly symmetric and a normal curve with mean lambda and standard deviation sqrt(lambda) approximates it well. For small lambda the distribution is strongly right-skewed and the normal approximation misstates tail probabilities. This calculator gives the exact values, so no approximation is needed.
How do I find the probability of at least k events?
Read the 'at least' card: P(X >= k) = 1 - P(X <= k - 1), computed from the upper tail so that a tiny value keeps its digits. For a range such as 3 to 8 events, switch to 'Between two values'; both limits are included.
How large can lambda be?
Up to 100,000,000 events per window. Within that range the probabilities have been checked against high-precision arithmetic to at least nine significant digits; larger rates are refused with a message rather than answered with a less accurate number.
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