Module 3 · Distributions · Topic 5
Poisson Distribution. The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen independently at a constant average rate λ.
Properties. For a Poisson distribution, both the mean and variance equal λ. This mean-equals-variance property is a distinctive feature.
P(X = k) = exp(−λ) × λᵏ / k!
P(X = k) = C(n,k)p^k(1-p)^(n-k) — Exact probability of k successes in n independent Bernoulli trials.z = (x - mu) / sigma — Standardizes a value by measuring standard deviations from the mean.A call center receives an average of 4 calls per minute (λ = 4). P(exactly 2 calls) = (e⁻⁴ × 4²) / 2! = (0.0183 × 16) / 2 = 0.1465.
The Poisson can approximate the binomial when n is large and p is small (np is moderate). Use λ = np as the rate parameter.
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