For a Poisson distribution with rate λ, what is the expected value E[X]?

For a Poisson distribution with rate λ, what is the expected value E[X]?

• A. 0
• B. 1/λ
• C. λ^2
• D. λ

Answer: (D)

The expected value of a Poisson random variable with rate λ is λ. This reflects the idea that in a fixed interval, events occur on average at rate λ, so the long-run average count is λ. You can see this from the formula E[X] = sum k·P(X=k) with P(X=k) = e^{-λ} λ^k / k!, and the sum evaluates to λ. A helpful intuition comes from the binomial-to-Poisson limit: if you have many trials with success probability p, and np = λ stays fixed while n grows large, the binomial mean np approaches λ, and the Poisson distribution inherits that same mean. Additionally, the variance of a Poisson(λ) is also λ, reinforcing that the parameter λ serves as both the average and the dispersion of X.

The expected value of a Poisson random variable with rate λ is λ. This reflects the idea that in a fixed interval, events occur on average at rate λ, so the long-run average count is λ. You can see this from the formula E[X] = sum k·P(X=k) with P(X=k) = e^{-λ} λ^k / k!, and the sum evaluates to λ. A helpful intuition comes from the binomial-to-Poisson limit: if you have many trials with success probability p, and np = λ stays fixed while n grows large, the binomial mean np approaches λ, and the Poisson distribution inherits that same mean. Additionally, the variance of a Poisson(λ) is also λ, reinforcing that the parameter λ serves as both the average and the dispersion of X.