Mean plus variance of an hour's total service time

Question

Customers reach a gift shop as a Poisson process averaging $7$ per hour, and serving each one takes exactly $10$ minutes. Measuring time in hours, let $T$ be the total service time needed for all customers who arrive during one particular hour. Find $\mathbb{E}[T]+\operatorname{Var}(T)$ .

Short answer

$\tfrac{49}{36}$ . The number of arrivals $N$ is $\text{Poisson}(7)$ and $T=N/6$ hours. Hence $\mathbb{E}[T]=\tfrac76$ and $\operatorname{Var}(T)=\tfrac{7}{36}$ , which add to $\tfrac{49}{36}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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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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