What is the Poisson distribution?

A discrete probability distribution that models the number of events occurring in a fixed interval, when events happen at a constant average rate independently of time since the last event.

What is lambda (λ) in Poisson distribution?

Lambda is the average rate parameter—the expected number of events per interval. Both the mean and variance of a Poisson distribution equal λ.

When should I use Poisson distribution?

For counting rare events in fixed intervals (time, area, volume) where events are independent and the average rate is known. Examples: calls per hour, defects per item, accidents per day.

What is the Poisson formula?

P(X = k) = (e^(-λ) · λ^k) / k! gives the probability of exactly k events when the average rate is λ.

Poisson Distribution Calculator

Calculate probabilities for Poisson distributions. Find P(X=k), cumulative probabilities, and visualize the distribution.

Rate Parameter (λ)

Average number of events per interval

Number of Events (x)

Specific number of events to calculate probability for

Formula: P(X = k) = (λ^k × e^(-λ)) / k!

About Poisson Distribution

When to Use

•Counting events in a fixed interval (time, area, volume)
•Events occur independently of each other
•Average rate of occurrence is constant
•Two events cannot occur at the exact same instant

Real-World Examples

•Number of calls at a call center per hour
•Website visitors per minute
•Defects in a manufacturing batch
•Car accidents at an intersection per month

Key Properties

Mean = Variance:E[X] = Var(X) = λ

Standard Deviation:σ = √λ

Skewness:1/√λ (right-skewed)

Mode:⌊λ⌋ or ⌊λ⌋-1

Related Math Tools

Compound Interest

Exponential growth

Note: The Poisson distribution assumes events are independent and occur at a constant average rate. For large λ (> 20), the normal approximation may be used.

About Poisson Distribution Calculator - Probability & Statistics

Calculate Poisson distribution probabilities for rare events. Find P(X=k), expected value, variance, and visualize the distribution.

Our **Poisson Distribution Calculator** computes probabilities for rare event occurrences. Enter the average rate (λ) and value of interest (k) to find P(X=k), cumulative probabilities, and distribution statistics. For other probability calculations, try our Twins Probability Calculator.

The Poisson distribution models events occurring independently at a constant average rate. Applications include call center arrivals, website clicks, radioactive decay, and error rates. Our calculator provides exact probabilities and helpful visualizations.

Beyond single probabilities, explore cumulative distribution functions, expected value (λ), variance (λ), and standard deviation (√λ). The interactive graph helps visualize probability mass for different parameter values.

Key Features

Exact probability P(X=k)

Cumulative probabilities

Distribution visualization

Mean and variance

Multiple k values

Probability tables

Why Use This Tool?

Quality control analysis

Risk assessment

Queue modeling

Scientific research

Statistical education

Common Use Cases

Call Centers: Probability of receiving k calls per hour.

Quality Control: Expected defects per batch.

Healthcare: Disease occurrence rates.

Insurance: Claims frequency modeling.

Related Tools

Geometric Mean Calculator

Average calculations.

Sum of Squares Calculator

Statistical analysis.

How to Use

Enter lambda (λ) - the average rate

Enter k - the number of events

Click Calculate

View probability and statistics