Poisson Distribution Calculator
Calculate Poisson probabilities for rare events. Find P(X = k), P(X ≤ k), and P(X > k) given an average rate λ.
Did this tool work for you?
How to use this calculator
λ is the average number of events in the interval. k is the specific count of interest. e is Euler's number (~2.71828).
- 1
Enter λ (lambda): the average number of events per interval (e.g. 3 calls per minute).
- 2
Enter k: the specific count you want the probability for (e.g. exactly 4 calls).
- 3
The calculator returns P(X = k), the cumulative P(X ≤ k), and P(X > k).
Frequently asked questions
What is the Poisson distribution used for?
The Poisson distribution models the number of independent events that occur in a fixed interval of time or space, when the average rate is known. Examples: call centre arrivals per minute, defects per product, accidents per year.
When does the Poisson distribution apply?
Use it when: events are independent; the rate is constant; two events cannot occur at exactly the same instant; and you're counting discrete events in a fixed interval.
How is Poisson different from binomial?
The binomial distribution models a fixed number of trials with probability p. The Poisson distribution models an unlimited number of possibilities with a small individual probability — it is the limiting case of the binomial as n → ∞ and p → 0 with λ = np fixed.
Poisson Distribution Calculator — P(X=k) & Cumulative Probability
Understanding the Poisson distribution
Named after Siméon Denis Poisson, this discrete probability distribution predicts the number of events in a fixed period when events happen at a constant average rate and independently of each other. Classic applications include queuing theory, insurance modelling, nuclear decay, and epidemiology.
Mean and variance of Poisson
A distinctive property of the Poisson distribution is that its mean and variance are both equal to λ. This makes it easy to estimate λ from data: just compute the sample mean. If the sample variance is much larger than the mean (overdispersion), a negative binomial distribution may be more appropriate.
Learn more from an authoritative source:
WikipediaVariance Calculator
Calculate variance and standard deviation for a set of numbers. Choose population variance (σ²) or sample variance (s²).
Standard Error Calculator
Calculate the Standard Error of the Mean (SEM) from standard deviation and sample size. Includes margin of error for 90%, 95%, and 99% confidence levels.
Correlation Coefficient Calculator
Calculate Pearson correlation coefficient (r) between two variables. Measures strength and direction of linear relationships.
Linear Regression Calculator
Calculate the linear regression line (y = mx + b) for a dataset. Find slope, intercept, and predict Y for any X value.
Z-Score Calculator
Calculate the z-score (standard score) and percentile rank for any value in a normal distribution. Convert raw scores to z-scores instantly.
T-Test Calculator
Perform a one-sample t-test to determine if a sample mean is significantly different from a known population mean. Returns t-statistic and p-value.
Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →