Answer¶
Let \(X\) be the waiting time and \(F_{X}(x)\) be the CDF. Then,
\[
\begin{aligned}
F_{X}(x) = \begin{cases} 0 &\mbox{ $x < 0$}\newline
\frac{2}{3} &\mbox{ $x = 0$}\newline
\frac{2}{3} + \frac{1}{30}x &\mbox{ $0 < x < 5$}\newline
1 &\mbox{ $5 \leq x$} \end{cases}
\end{aligned}
\]
The PDF is simply the derivate of the CDF. Thus, expectation is
\[
\begin{aligned}
E[X] = \frac{2}{3}(0) + \int_{0}^{5} \frac{1}{30}x dx + \frac{1}{6}(5) = \frac{5}{4} mins
\end{aligned}
\]