Calculate exact and cumulative Poisson probabilities given an average event rate λ. Explore the distribution with interactive charts and key statistics.
Enter the average event rate (λ) and a target value to explore the Poisson distribution and calculate probabilities.
The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, when these events occur with a known constant mean rate and independently of the time since the last event. It's named after French mathematician Siméon Denis Poisson and is one of the most important discrete probability distributions.
Where e is Euler's number (approximately 2.71828), λ is the average rate, and k! is the factorial of k. This formula gives the probability of observing exactly k events when the average rate is λ.
μ = λ
The expected number of events equals the rate parameter.
σ² = λ
Uniquely, variance equals the mean in Poisson distributions.
σ = √λ
The spread of the distribution grows with √λ.
The Poisson distribution is valid only when these conditions are met:
From Binomial: The Poisson distribution is a limiting case of the binomial distribution when n is large, p is small, and λ = n × p is moderate. This is called the "Law of Rare Events."
To Normal: For large λ (typically λ > 20), the Poisson distribution can be approximated by a normal distribution with μ = λ and σ = √λ.
Exponential Connection: If events follow a Poisson process with rate λ, the time between consecutive events follows an exponential distribution.
Exact Probability P(X = k): The probability of observing exactly k events. Use this for questions like "What's the chance of exactly 5 calls in an hour?"
Cumulative P(X ≤ k): The probability of observing k or fewer events. Useful for questions like "What's the chance of at most 3 defects per batch?"
Cumulative P(X ≥ k): The probability of observing k or more events. Useful for questions like "What's the chance of at least 10 customers per hour?"
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