Variance of library return processing time

Question

A library system logs a returned book in $2$ hours if it comes back to its home branch, and in $5$ hours if it is dropped at a different branch. $70\%$ of borrowers use their home branch. Find the variance of the logging time.

Short answer

$1.89$ . Write the time as $2+3I$ , where $I$ indicates a return to another branch, a Bernoulli $(0.3)$ variable with variance $0.21$ . The variance is then $9\times0.21=1.89$ hours $^2$ .

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Covariance and Correlation

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