Which statement best describes when the Poisson distribution is typically used?

• A. The Poisson distribution models rare events in a fixed interval.
• B. The Normal distribution models discrete data.
• C. The Binomial distribution models continuous data.
• D. The Uniform distribution models independent events.

Answer: (A)

This is about when the Poisson model fits a situation. It applies best when you’re counting how many times a rare event occurs within a fixed interval of time or space, and those events happen with a roughly constant average rate and independently from one another. The Poisson distribution uses a single parameter λ, the average number of events in that interval, to describe the probabilities of different counts: P(X = k) = e^(-λ) λ^k / k!. This makes it ideal for cases like the number of emails arriving per hour, the number of cars crossing a bridge in a minute, or any scenario where events are rare enough that occurrences in one part of the interval don’t strongly affect another.

The other options describe different kinds of models and aren’t about counting rare events in a fixed interval. The Normal distribution is for continuous data, often arising as a good approximation when many small effects add up. The Binomial distribution counts successes in a fixed number of trials and does not describe counts within a continuous interval; it’s about fixed trials with two outcomes. The Uniform distribution assigns equal probability to a finite set of outcomes, not about how often events occur over time.