Probability
Poisson Distribution Calculator
Compute the Poisson probability of a count, the cumulative probability up to that count, and the mean, which equals the variance.
Worksheet
Press Calculate to see the result.
Compute the Poisson probability of a count, the cumulative probability up to that count, and the mean, which equals the variance.
What this calculator answers
A Poisson count is a model for events that arrive independently at a steady average rate, with no fixed number of trials. Calls in an hour, defects on a length of wire, and rare accidents in a month are the usual pictures. You supply the average count lambda and the count k you care about. The page returns the probability of that exact count, the probability of k or fewer, and it reminds you that the mean and the variance are the same number.
The point probability and the cumulative probability come from the Poisson routines. The mean and the variance are both set equal to lambda. There is no separate variance input. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Poisson probability mass uses this relationship.
P(X = k) = e^(−λ) λ^k / k!, mean = λ, variance = λ
- λ. The average count in the interval, also called lambda.
- k. The observed count, a whole number starting at 0.
- e^(−λ). The probability of zero events.
- k!. The factorial that normalizes the mass at k.
Independent arrivals at a constant rate produce counts whose probabilities are proportional to lambda to the k, damped by the exponential of minus lambda, and divided by k factorial so the masses add to 1. Differentiating the probability generating function shows that both the mean and the variance equal lambda. That equality is the signature of the model. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
Lambda is 3 and the count k is 2. The Poisson point probability P(X = 2) is 0.224. The cumulative probability P(X ≤ 2) is 0.4232.
The mean of this distribution is lambda, so the mean is 3. The variance of a Poisson distribution is also lambda, so the variance is 3. Those two results match on purpose. A Poisson model does not have a separate variance parameter.
k = 0 would have been allowed. A negative count or a lambda of 0 is not. This is a count in an interval, not a fixed number of trials. A fixed n belongs on the binomial page.
| P(X = k) | 0.224 |
|---|---|
| P(X ≤ k) | 0.4232 |
| Mean | 3 |
| Variance | 3 |
A second situation
A higher mean with the same count
Lambda is 8 and the count k is 2. The Poisson point probability P(X = 2) is 0.0107. The cumulative probability P(X ≤ 2) is 0.0138.
The mean of this distribution is lambda, so the mean is 8. The variance of a Poisson distribution is also lambda, so the variance is 8. Those two results match on purpose. A Poisson model does not have a separate variance parameter.
k = 0 would have been allowed. A negative count or a lambda of 0 is not. This is a count in an interval, not a fixed number of trials. A fixed n belongs on the binomial page.
| P(X = k) | 0.0107 |
|---|---|
| P(X ≤ k) | 0.0138 |
| Mean | 8 |
| Variance | 8 |
Raising lambda from 3 to 8 while holding k at 2 moves the count from near the mean to well below it, so the point probability drops. The variance rises with lambda even though you never typed a variance. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The count moved from 2 to 5 while lambda stayed at 3. The headline result went from 0.224 to 0.1008.
The nudged inputs
Lambda is 3 and the count k is 5. The Poisson point probability P(X = 5) is 0.1008. The cumulative probability P(X ≤ 5) is 0.9161.
The mean of this distribution is lambda, so the mean is 3. The variance of a Poisson distribution is also lambda, so the variance is 3. Those two results match on purpose. A Poisson model does not have a separate variance parameter.
k = 0 would have been allowed. A negative count or a lambda of 0 is not. This is a count in an interval, not a fixed number of trials. A fixed n belongs on the binomial page.
| P(X = k) | 0.1008 |
|---|---|
| P(X ≤ k) | 0.9161 |
| Mean | 3 |
| Variance | 3 |
How to read the result
Read the point probability as the chance of exactly k events, and the cdf as the chance of at most k. If the variance in real data is much larger than the mean, the Poisson headline is the wrong model even though the arithmetic ran. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- Events are counted in a fixed interval of time, space, or exposure.
- The average rate is constant across that interval.
- Arrivals are independent.
- Two events do not pile onto the exact same instant in a way that creates extra clustering.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Typing a variance that differs from lambda and expecting the page to use it. This model refuses that freedom.
- Using a Poisson model for a fixed number of trials. That is binomial.
- Forgetting that k = 0 is allowed. Zero events has probability e to the minus lambda.
- Entering a negative lambda. A mean count cannot be negative, and a zero rate is rejected because the mass is degenerate.
Where a different calculator is the honest choice
Use the binomial calculator when the number of trials is fixed. Use the exponential calculator for the waiting time between Poisson events, which is a different random variable. Use the geometric calculator when you are counting trials until the first success rather than events in an interval. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Binomial Distribution Calculator, Exponential Distribution Calculator, Geometric Distribution Calculator.
Check yourself
If the mean count is 3, what is the probability of exactly 2 events, and what is the variance?
P(X = 2) is 0.224 and P(X ≤ 2) is 0.4232. The mean is 3 and the variance is also 3.
Using the result in a sentence
Calls per hour, insurance claims per month, and defects per batch are classic Poisson questions when the rate is stable. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Overdispersion is common in real counts. If the sample variance is far from the sample mean, a Poisson probability will look more precise than the process deserves. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Poisson Distribution Calculator commit to?
This page uses Poisson probability mass: P(X = k) = e^(−λ) λ^k / k!, mean = λ, variance = λ. Independent arrivals at a constant rate produce counts whose probabilities are proportional to lambda to the k, damped by the exponential of minus lambda, and divided by k factorial so the masses add to 1. Differentiating the probability generating function shows that both the mean and the variance equal lambda. That equality is the signature of the model. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Mean lambda, Count k, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the p(x = k) is 0.224. Read the point probability as the chance of exactly k events, and the cdf as the chance of at most k. If the variance in real data is much larger than the mean, the Poisson headline is the wrong model even though the arithmetic ran. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Typing a variance that differs from lambda and expecting the page to use it. This model refuses that freedom. Overdispersion is common in real counts. If the sample variance is far from the sample mean, a Poisson probability will look more precise than the process deserves. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the binomial calculator when the number of trials is fixed. Use the exponential calculator for the waiting time between Poisson events, which is a different random variable. Use the geometric calculator when you are counting trials until the first success rather than events in an interval. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
P(X = 2) is 0.224 and P(X ≤ 2) is 0.4232. The mean is 3 and the variance is also 3. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.