How to use the poisson distribution calculator
- Enter lambda, the expected average number of events in your interval.
- Enter k, the specific number of events you want to evaluate.
- Read the results:
P(X = k)is the exact probability, whileP(X ≤ k)andP(X ≥ k)give cumulative and tail probabilities. - Check the mean and variance rows, which both equal lambda for a Poisson model.
How the Poisson formula works
The Poisson distribution gives the probability of observing exactly k events when the average rate is lambda. The probability mass function is:
P(X = k) = (e-λ · λk) / k!
For example, if a call center receives an average of 3 calls per hour (λ = 3) and you want the probability of exactly 2 calls (k = 2):
P(X = 2) = e-3 · 32 / 2! = 0.049787 · 9 / 2 = 0.224042
The cumulative probability P(X ≤ k) is the sum of P(X = i) for all i from 0 to k. The tail probability P(X ≥ k) equals 1 minus P(X ≤ k-1). Both the mean and the variance of a Poisson distribution are exactly equal to lambda.
Reference table for lambda = 4
This table shows exact and cumulative probabilities when the average rate is 4 events per interval. You can verify these values using the poisson distribution calculator above.
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.018316 | 0.018316 |
| 1 | 0.073263 | 0.091579 |
| 2 | 0.146525 | 0.238104 |
| 3 | 0.195367 | 0.433471 |
| 4 | 0.195367 | 0.628838 |
| 5 | 0.156293 | 0.785131 |
| 6 | 0.104196 | 0.889327 |
Practical use cases
The Poisson distribution applies whenever you count discrete events that occur independently at a constant average rate. Common examples include:
- Customer arrivals: modeling how many customers enter a store per hour when the average is known.
- Network traffic: counting packet arrivals or server requests per second.
- Quality control: estimating the number of defects per manufactured batch.
- Safety analysis: predicting accident counts per month on a stretch of road.
In each case, lambda represents the historical average, and k is the scenario you want to evaluate. If you need to convert betting odds into event likelihoods, an implied probability calculator handles that conversion separately.
Common mistakes and edge cases
Several pitfalls arise when applying the Poisson model:
- Non-integer k: the Poisson distribution only applies to whole numbers of events. Passing k = 2.5 is invalid because you cannot observe half an event.
- Negative lambda: the rate parameter must be zero or positive. A negative average rate has no physical meaning.
- Dependent events: if events influence each other (for example, one customer attracting another), the independence assumption breaks down and the Poisson model becomes inaccurate.
- Varying rates: lambda must remain constant over the interval. If the rate changes (rush hour versus off-peak), split the problem into sub-intervals with separate lambdas.
- Large k values: computing k! directly causes overflow for large k. Reliable implementations use logarithms or iterative multiplication to keep intermediate values manageable.
Relationship to other distributions
The Poisson distribution connects to several other statistical models. It serves as a limiting case of the binomial distribution when the number of trials n grows large and the success probability p shrinks so that n · p remains constant at lambda. In practice, if n exceeds 20 and p is below 0.05, the Poisson approximation to the binomial is quite accurate.
The waiting time between consecutive Poisson events follows an exponential distribution with rate parameter lambda. If you count the time until the r-th event instead, the result follows a gamma distribution. These relationships make the Poisson process a foundational building block in queueing theory, reliability engineering and stochastic modeling.
Worked examples
Each result below was produced by the same code the tool runs, so you can check your own numbers against them.
| Example | Input | Result |
|---|---|---|
| Call center with 5 calls per hour, exactly 3 | Lambda (average rate): 5; k (number of events): 3 | P(X = 3): 0.140374; P(X ≤ 3): 0.265026; P(X ≥ 3): 0.875348; Mean: 5; Variance: 5 |
| Website errors averaging 2 per day, exactly 0 | Lambda (average rate): 2; k (number of events): 0 | P(X = 0): 0.135335; P(X ≤ 0): 0.135335; P(X ≥ 0): 1; Mean: 2; Variance: 2 |
| Defects at 0.5 per batch, exactly 1 | Lambda (average rate): 0.5; k (number of events): 1 | P(X = 1): 0.303265; P(X ≤ 1): 0.909796; P(X ≥ 1): 0.393469; Mean: 0.5; Variance: 0.5 |
| Large lambda: 50 events, exactly 50 | Lambda (average rate): 50; k (number of events): 50 | P(X = 50): 0.056325; P(X ≤ 50): 0.537517; P(X ≥ 50): 0.518808; Mean: 50; Variance: 50 |
FAQ
What does lambda represent in the Poisson distribution?
Lambda is the expected average number of events in a fixed interval of time or space. For instance, if a website averages 5 errors per day, lambda equals 5.
Can k be a decimal number?
No. The variable k represents a count of discrete events, so it must be a non-negative integer such as 0, 1, 2 or 3.
Why are the mean and variance identical?
A defining property of the Poisson distribution is that both the mean and the variance equal lambda. If your data shows a variance much larger than the mean, the Poisson model may not fit well.
What is the difference between P(X=k) and P(X<=k)?
P(X=k) is the probability of observing exactly k events. P(X<=k) is the cumulative probability of observing k or fewer events, computed by summing P(X=i) for all i from 0 to k.
When should I not use the Poisson distribution?
Avoid it when events are not independent, when the average rate changes during the observation period, or when the variance in your data differs substantially from the mean.
Does this tool send my data to a server?
No. All calculations happen locally in your browser. Nothing is uploaded or stored.