📊 Poisson Distribution Calculator
Calculate P(X=k), P(X≤k), and P(X>k) for a Poisson distribution. Visualise the probability mass function as a bar chart.
Probability Bar Chart
Poisson Distribution Formulas
Common Applications of Poisson Distribution
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1Call Centre ModellingEstimating the probability of receiving exactly k calls per hour when the average rate is λ calls/hour.
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2Quality ControlModelling the number of defects per unit of product when defects occur rarely and independently.
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3Medical EventsPredicting the number of hospital admissions per day for rare conditions.
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4Traffic AnalysisModelling the number of vehicles passing a checkpoint per minute.
Frequently Asked Questions
The Poisson distribution models the number of events occurring in a fixed interval of time or space, given that events occur independently at a constant average rate λ. It is used for rare events.
Lambda is the average number of events per interval. It is both the mean and variance of the Poisson distribution. A larger λ shifts the distribution to the right and makes it more symmetric.
Use Poisson when n is large and p is small (rare events), such that np ≈ λ. The Poisson distribution approximates the binomial when n > 20 and p < 0.05.
The cumulative probability P(X ≤ k) is the sum of P(X = 0) + P(X = 1) + ... + P(X = k). It tells you the probability of observing at most k events.
Yes. The Poisson distribution is a discrete probability distribution — k must be a non-negative integer (0, 1, 2, ...). There is no upper limit on k, though probabilities become negligible after a few multiples of λ.
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