Poisson distribution models count data observed in a fixed interval. What is the interpretation of its parameter λ?

Poisson distribution models count data observed in a fixed interval. What is the interpretation of its parameter λ?

• A. λ is the standard deviation of the counts.
• B. λ is the mean and the variance of the distribution.
• C. λ is the probability of success in a Bernoulli trial.
• D. λ is the rate parameter for exponential waiting times.

Answer: (B)

In a Poisson model, the parameter λ represents the average rate of events per fixed interval and also the expected count in that interval. Concretely, if X is the number of events in the interval, then E[X] = λ and Var(X) = λ. That means λ is both the mean and the variance of the distribution. The standard deviation would be sqrt(λ), not λ itself. So λ tells you how many events you expect on average in that interval, and it also determines how spread out those counts are around that average. This is distinct from a Bernoulli probability p or from parameters used for exponential waiting times, though the exponential waiting time distribution with rate λ is connected to the Poisson process: the times between events are exponential with the same rate λ, while the Poisson counts in an interval have mean and variance equal to λ.

In a Poisson model, the parameter λ represents the average rate of events per fixed interval and also the expected count in that interval. Concretely, if X is the number of events in the interval, then E[X] = λ and Var(X) = λ. That means λ is both the mean and the variance of the distribution. The standard deviation would be sqrt(λ), not λ itself.

So λ tells you how many events you expect on average in that interval, and it also determines how spread out those counts are around that average. This is distinct from a Bernoulli probability p or from parameters used for exponential waiting times, though the exponential waiting time distribution with rate λ is connected to the Poisson process: the times between events are exponential with the same rate λ, while the Poisson counts in an interval have mean and variance equal to λ.