Recognize distribution moments

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Problem

Give the mean and variance of $\operatorname{Binomial}(n,p)$ , $\operatorname{Poisson}(\lambda)$ , $\operatorname{Exponential}(\lambda)$ under the rate parameterization, and $\mathcal{N}(\mu,\sigma^2)$ . Then identify the distribution of a sum of independent Poisson variables.

Reveal answer or reference solution

$\operatorname{Binomial}(n,p)$ : mean $np$ and variance $np(1-p)$ .
$\operatorname{Poisson}(\lambda)$ : mean $\lambda$ and variance $\lambda$ .
$\operatorname{Exponential}(\lambda)$ : mean $1/\lambda$ and variance $1/\lambda^2$ .
$\mathcal{N}(\mu,\sigma^2)$ : mean $\mu$ and variance $\sigma^2$ .

Independent Poisson variables sum to a Poisson variable with rate equal to the sum of the rates.

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Appears in

Probability