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

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

• A. λ/2
• B. λ
• C. λ^2
• D. 2λ

Answer: (B)

For a Poisson random variable, the spread around the average count is governed by the same parameter that sets the average itself. Specifically, if X follows a Poisson distribution with rate λ, then the mean is λ and the variance is also λ. A quick way to see this is to use the identities E[X] = λ and E[X(X−1)] = λ^2 for a Poisson(λ). Since E[X^2] = E[X(X−1)] + E[X], we get E[X^2] = λ^2 + λ. The variance is Var(X) = E[X^2] − (E[X])^2 = (λ^2 + λ) − λ^2 = λ. So the number of events in a fixed interval not only averages λ but also fluctuates with a variance of λ, reflecting that larger λ means both higher average and greater dispersion.

For a Poisson random variable, the spread around the average count is governed by the same parameter that sets the average itself. Specifically, if X follows a Poisson distribution with rate λ, then the mean is λ and the variance is also λ.

A quick way to see this is to use the identities E[X] = λ and E[X(X−1)] = λ^2 for a Poisson(λ). Since E[X^2] = E[X(X−1)] + E[X], we get E[X^2] = λ^2 + λ. The variance is Var(X) = E[X^2] − (E[X])^2 = (λ^2 + λ) − λ^2 = λ.

So the number of events in a fixed interval not only averages λ but also fluctuates with a variance of λ, reflecting that larger λ means both higher average and greater dispersion.