[Exer14-4] Exercise 7 of Book {Devore2017B} P.212
The joint probability distribution of the number $X$ of cars and the number $Y$ of buses per signal cycle at a proposed left turn lane is displayed in the accompanying joint probability table.
| $y$ | ||||
| $p(x,y)$ | 0 | 1 | 2 | |
| 0 | .025 | .015 | .010 | |
| 1 | .050 | .030 | .020 | |
| $x$ | 2 | .125 | .075 | .050 |
| 3 | .150 | .090 | .060 | |
| 4 | .100 | .060 | .040 | |
| 5 | .050 | .030 | .020 | |
%%Table in LaTeX
\begin{table}[htbp]
\centering
\begin{tabular}{cc|p{0.5in}p{0.5in}p{0.5in}}
& & & $y$ & \\
$p(x,y)$ & & 0 & 1 & 2 \\\hline
\multirow{3}{*}{$x$}& 0 & .025 & .015 & .010\\
~& 1 & .050 & .030 & .020\\
~& 2 & .125 & .075 & .050\\
~& 3 & .150 & .090 & .060\\
~& 4 & .100 & .060 & .040\\
~& 5 & .050 & .030 & .020\\
\hline
\end{tabular}
\end{table}
- (a) What is the probability that there is exactly one car and exactly one bus during a cycle?
- (b) What is the probability that there is at most one car and at most one bus during a cycle?
- (c) What is the probability that there is exactly one car during a cycle? Exactly one bus?
- (d) Suppose the left turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflow during a cycle?
- (e) Are $X$ and $Y$ independent rv's? Explain.