For a Poisson distribution, which statement about λ is correct?

For a Poisson distribution, which statement about λ is correct?

• A. λ is the rate parameter for the Bernoulli distribution.
• B. λ is the mean of the distribution.
• C. λ is the standard deviation.
• D. λ equals both the mean and the variance.

Answer: (D)

In a Poisson model, the parameter λ represents the average rate of occurrences, and the distribution has both its mean and its variance equal to λ. Specifically, if X ~ Poisson(λ), then E[X] = λ and Var(X) = λ, so the standard deviation is sqrt(λ). Because of these two tied properties, a statement that λ equals both the mean and the variance captures the full role of λ in a Poisson distribution. The other options mix up concepts: the rate parameter for a Bernoulli distribution uses p, not λ; the standard deviation is not λ but sqrt(λ); and saying λ is the mean alone is true but incomplete compared to the complete fact that it also equals the variance.

In a Poisson model, the parameter λ represents the average rate of occurrences, and the distribution has both its mean and its variance equal to λ. Specifically, if X ~ Poisson(λ), then E[X] = λ and Var(X) = λ, so the standard deviation is sqrt(λ). Because of these two tied properties, a statement that λ equals both the mean and the variance captures the full role of λ in a Poisson distribution. The other options mix up concepts: the rate parameter for a Bernoulli distribution uses p, not λ; the standard deviation is not λ but sqrt(λ); and saying λ is the mean alone is true but incomplete compared to the complete fact that it also equals the variance.