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The Poisson_Dist command illustrates the properties of the Poisson distribution through a set of interactive charts. The Poisson distribution is commonly used to model the number of events occurring within a fixed interval of time or space when events occur independently and at a constant average rate.
Main idea: provide an intuitive visual understanding of one of the most important discrete probability distributions used in statistics, finance, insurance, engineering, and operations research.
Poisson_Dist(5)
This example creates a Poisson distribution with an expected arrival rate of 5 events per interval.
Poisson_Dist(lambda)
where:
lambda – the expected number of events occurring during the observation period.If a random variable X follows a Poisson distribution with parameter λ, then the probability of observing exactly k events is:
P(X = k) = e-λ λk / k!
for k = 0, 1, 2, ...
The mean and variance of the Poisson distribution are both equal to λ.
The Poisson distribution is often used to model rare or random events such as:
As λ increases, the distribution becomes more symmetric and begins to resemble a normal distribution. For small values of λ, the distribution is highly skewed, reflecting the rarity of events.