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Poisson Distribution Formula:
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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.
The calculator uses the Poisson formula:
Where:
Explanation: The formula calculates the probability of exactly k events occurring in a fixed interval when the events occur independently and at a constant average rate λ.
Details: The Poisson distribution is widely used in various fields including telecommunications, astronomy, biology, and business to model random events such as call arrivals, radioactive decay, website visits, and customer arrivals.
Tips: Enter lambda (average rate) as a positive number and k (number of events) as a non-negative integer. Both values must be valid (λ > 0, k ≥ 0).
Q1: When should I use the Poisson distribution?
A: Use it when events occur independently, at a constant average rate, and you want to find the probability of a certain number of events in a fixed interval.
Q2: What's the relationship between Poisson and binomial distributions?
A: The Poisson distribution can be derived as a limiting case of the binomial distribution when the number of trials is large and the probability of success is small.
Q3: What are typical applications of Poisson distribution?
A: It's used in queuing theory, insurance risk assessment, reliability engineering, and predicting rare events like natural disasters.
Q4: Are there limitations to the Poisson distribution?
A: It assumes events are independent and the average rate is constant. It may not be appropriate if events tend to cluster or have seasonal patterns.
Q5: What does the lambda parameter represent?
A: Lambda (λ) represents the average number of events in the given time interval or spatial region.