Calculate PMF, CDF, mean and variance for the Poisson distribution with rate λ.
The Poisson distribution models the number of events occurring in a fixed interval when events happen at a constant average rate λ and independently of each other. Both its mean and variance equal λ.
P(X=k) = (λ^k × e^(−λ)) / k!
When counting rare events in a fixed interval (time, area, volume) where events occur independently at a constant average rate — e.g., calls per hour, defects per batch.
The Poisson is the limit of the binomial as n→∞ and p→0 while np→λ. It approximates the binomial well when n is large and p is small.