Solutions - 5240-03

(1) Define random variables to describe the problem:

Let $X\sim \mathrm{Unif}[0,120]$ represent the arrival time of the plumber. Let $Y\sim \text{Exp}\left(\frac{1}{45}\right)$ represent the completion time of the sink fix.

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(2) Compute $E[X+Y]$:

This represents the expected time the plumber finishes the job.

$$\begin{array}{c}E[X+Y]=E[X]+E[Y]=60+45\gg \gg 105\end{array}$$

Thus, we expect the plumber to finish at $3:45$.

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(3) Compute the variance of the finish time:

$$\begin{array}{c}\text{Var}[X+Y]=\text{Var}[X]+\text{Var}[Y]+2\text{Cov}[X,Y]=\frac{{120}^2}{12}+\frac{1}{{\left(\frac{1}{45}\right)}^2}\gg \gg 3225{\text{ min}}^2\end{array}$$

As part of the problem interpretation, we assume that the time to fix a sink is independent of the starting time, so $\mathrm{Cov}[X,Y]=0$.