Poisson Distribution Probability Calculator
Calculate probabilities for rare events occurring in a fixed interval. Generates PMF and CDF values given Lambda (λ) and k.
About
The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, assuming these events occur with a known constant mean rate and independently of the time since the last event. It is widely used in fields ranging from telecommunications (call arrival rates) to biology (mutation counts).
This tool computes the Probability Mass Function (PMF), which is the probability of observing exactly k events, and the Cumulative Distribution Function (CDF). It is essential for operations research, inventory management, and risk assessment where discrete, rare events are analyzed.
Formulas
The probability of observing exactly k events is given by the formula:
P(k) = λk e−λk!
Where:
- e is Euler's number (≈ 2.71828)
- k! is the factorial of k
The cumulative probability is the sum of probabilities from 0 to k.
Reference Data
| Variable | Symbol | Definition | Example |
|---|---|---|---|
| Mean Rate | λ (Lambda) | Average number of events per interval. | 5 calls / hour |
| Occurrences | k | Actual number of successes to test. | Exactly 3 calls |
| PMF | P(X=k) | Prob. of exactly k events. | 0.1404 |
| CDF | P(X≤k) | Prob. of k or fewer events. | 0.2650 |
| Complement | P(X>k) | Prob. of more than k events. | 0.7350 |