Poisson Distribution Calculator
This calculator computes Poisson distribution probabilities for events occurring in a fixed interval. Enter the average rate (λ) and the desired number of occurrences (x) to calculate probabilities.
Poisson Distribution Formula
The probability of observing exactly x events in an interval is given by:
P(X = x) = (e-λ * λx) / x!
Where:
- λ is the average rate of occurrence
- x is the number of occurrences
- e is Euler's number (~2.71828)
- x! is the factorial of x
Poisson Distribution Results
Enter the average rate (λ) and number of occurrences (x) to calculate probabilities.
| Average rate (λ) | - |
|---|---|
| Number of occurrences (x) | - |
| Probability Type | - |
| Probability | - |
| Mean (μ) | - |
| Variance (σ²) | - |
About the Poisson Distribution
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.
When to Use the Poisson Distribution
The Poisson distribution is appropriate for applications with the following characteristics:
- Events occur independently - the occurrence of one event does not affect the probability of another
- The average rate (λ) of events is constant
- Two events cannot occur at exactly the same instant
- The probability of an event in a small interval is proportional to the length of the interval
Examples of Poisson Processes
Common examples where the Poisson distribution applies include:
- Number of phone calls received by a call center per hour
- Number of decay events per second from a radioactive source
- Number of visitors to a website per minute
- Number of mutations in a given stretch of DNA after a certain amount of radiation
- Number of printing errors per page in a large document
Properties of the Poisson Distribution
The Poisson distribution has several important properties:
- Mean: E(X) = λ
- Variance: Var(X) = λ
- Standard Deviation: σ = √λ
- Skewness: 1/√λ (positive skew that decreases as λ increases)
- Kurtosis: 1/λ (excess kurtosis)
Relationship to Other Distributions
The Poisson distribution is related to several other probability distributions:
- Binomial: The Poisson distribution can be derived as a limiting case of the binomial distribution when n → ∞ and p → 0 while np remains fixed (np = λ)
- Normal: For large values of λ (typically λ > 20), the Poisson distribution can be approximated by a normal distribution with mean λ and variance λ
- Exponential: The time between events in a Poisson process follows an exponential distribution
Limitations
The Poisson distribution may not be appropriate when:
- Events are not independent (e.g., if one event makes subsequent events more or less likely)
- The average rate changes over time
- Multiple events can occur simultaneously