In a Poisson distribution with parameter λ, what is the mean?

In a Poisson distribution with parameter λ, what is the mean?

• A. λ^2
• B. λ
• C. 1
• D. 0

Answer: (B)

In a Poisson distribution, the average rate of events per interval is written as λ, and that rate is exactly the mean number of events we expect in that interval. So the expected value E[X] equals λ. This comes from the defining probability mass function or, equivalently, from viewing Poisson as the limit of a binomial model with n large and p small while np = λ. Intuitively, if events occur independently at a constant average rate λ, the long-run average count in an interval is λ. The other numbers would not match this average behavior: they would imply the mean grows in a way that isn’t tied to the rate λ. Hence the mean is λ.

In a Poisson distribution, the average rate of events per interval is written as λ, and that rate is exactly the mean number of events we expect in that interval. So the expected value E[X] equals λ. This comes from the defining probability mass function or, equivalently, from viewing Poisson as the limit of a binomial model with n large and p small while np = λ. Intuitively, if events occur independently at a constant average rate λ, the long-run average count in an interval is λ.

The other numbers would not match this average behavior: they would imply the mean grows in a way that isn’t tied to the rate λ. Hence the mean is λ.