Mean plus variance of rug repair cost

Question

Flaws in a handmade rug occur as a Poisson count with mean $4$ per square yard, and fixing each flaw costs $10. For a rug of area $10$ square yards, compute the expected repair cost plus the variance of the repair cost.

Short answer

$4400$ . The flaw count on 10 square yards is $N\sim\text{Poisson}(40)$ , so $\mathbb{E}[N]=\text{Var}(N)=40$ . The cost $10N$ has mean $400$ and variance $100\cdot40=4000$ , summing to $4400$ .

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Lessons behind this question

Defining the Poisson Process: Three Equivalent Constructions
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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