Expected rope profit with Poisson defects

Question

Each foot of rope has a number of defects $X\sim\text{Poisson}(2)$ , and the profit on that foot is $50-2X-X^2$ . Find the expected profit per foot.

Short answer

$40$ . For a Poisson(2) variable $E[X]=2$ and $\operatorname{Var}(X)=2$ , so $E[X^2]=2+4=6$ . The expected profit is $50-2(2)-6=40$ .

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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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