Calculate Poisson probability P(X=k), cumulative event odds, and arrival rates per time or spatial interval.
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The Poisson Distribution models the number of times an event occurs within a fixed interval of time or space. It applies when events happen with a known constant mean rate λ and independently of the time since the last event (e.g. call center calls per hour, website traffic bursts, emergency room admissions).
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad \mu = \lambda, \quad \sigma^2 = \lambdaWhere λ is the average number of events per interval, k is the observed number of events, and e is Euler’s constant (≈ 2.71828). In a Poisson distribution, the mean and variance are uniquely equal to λ.
Scenario: A coffee shop receives an average of 3 customers per minute (λ = 3). What is the probability of exactly 4 customers arriving in the next minute?
Parameters: λ = 3, k = 4.
Calculate numerator: 3^4 × e^(-3) = 81 × 0.049787 = 4.03275.
Calculate denominator: 4! = 24.
Divide: 4.03275 / 24 ≈ 0.1680 (16.80%).
Events must occur one at a time, events must be independent of one another, and the average rate λ must remain constant across the measured period.
Multiply the base rate by the time factor. If website errors occur at λ = 2 per hour, the rate for an 8-hour shift is λ = 2 × 8 = 16 errors.
If the count of events in an interval follows a Poisson distribution with rate λ, the elapsed time between successive events follows an exponential distribution with rate parameter λ.
Yes. When n is large (n ≥ 100) and p is small (p ≤ 0.01), a binomial distribution is well approximated by a Poisson distribution with λ = n × p.
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