Poisson distribution formula: P(k; λ) = (λ^k * e^(-λ)) / k!
Poisson distribution
Poisson distribution formula: P(k; λ) = (λ^k * e^(-λ)) / k!
The Poisson distribution formula calculates the probability of observing k events in a fixed interval when events occur independently at a constant mean rate λ. This formula is essential for understanding the likelihood of various outcomes in scenarios modeled by Poisson processes.
Example
If a bookstore averages 3 sales per hour (λ = 3), the probability of exactly 2 sales in the next hour (k = 2) is calculated using the Poisson formula: P(2; 3) = (3^2 * e^(-3)) / 2! = (9 * e^(-3)) / 2 ≈ 0.224.
Understanding the Poisson distribution formula is crucial for accurately predicting event probabilities in fields like telecommunications, insurance, and traffic flow management.
Related concepts
Expected value
Expected value formula: E[X] = Σ [x * P(x)]
Conditional probability
P(A|B) = P(A ∩ B) / P(B)
Bayes' theorem
Bayes' theorem formula: P(A|B) = [P(B|A) * P(A)] / P(B)
Normal distribution
Normal distribution PDF formula
Logistic regression
Logistic regression probability formula: P(Y=1) = 1 / (1 + exp(-z))
Entropy (information theory)
H(X) = −∑x∈X p(x) log(p(x))
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