This is a problem 8 from Chapter 8 "Introduction to probability" by Anderson:
Our faucet is broken, and a plumber has been called. The arrival time of the plumber is uniformly distributed between 1pm and 7pm. The time required to fix the broken faucet is then exponentially distributed with mean 30 minutes. Supposing that the two times are independent, find the expected value and the variance of the time at which the plumber completes the project.
I've solved it, but I am not sure if I interpreted the statement of the problem (translated into the language of probability distribution) correctly. So let X be the arrival time of the plumber, then $X\sim Unif[1,7]$. Let Y be the time it takes to fix the broken faucet, then $Y\sim Exp(2)$. Let Z be the time at which the plumber completes the project, then $Z=X+Y$. Now $E[Z]=E[X+Y]=E[X]+E[Y]=\frac{1+7}{2}+\frac{1}{2}=4.5$ $Var(Z) = \text{(since X and Y are independent)} \ Var(X)+Var(Y) = \frac{{(7-1)}^2}{12}+\frac{1}{2^2}=3.25$
Are there any mistakes in my solution? Thanks.
Looks good. You have identified the distributions -and their mean and variance- correctly, and used the right equations -including the effect of independance-. If the calculations are okay, then you have it.