[Exer14-1] Exercise 1 of Book {Devore2017B} P.211
A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let $X$ denote the number of hoses being used on the self-service island at a particular time, and let $Y$ denote the number of hoses on the full-service island in use at that time. The joint pmf of $X$ and $Y$ appears in the accompanying tabulation.
| $y$ | ||||
| $p(x,y)$ | 0 | 1 | 2 | |
| 0 | .10 | .04 | .02 | |
| $x$ | 1 | .08 | .20 | .06 |
| 2 | .06 | .14 | .30 | |
%%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 & .10 & .04 & .02\\
~& 1 & .08 & .20 & .06\\
~& 2 & .06 & .14 & .30\\
\hline
\end{tabular}
\end{table}
- (a) What is $P(X=1\ \text{and}\ Y=1)$?
- (b) Compute $P(X\leqslant 1\ \text{and}\ Y\leqslant 1)$.
- (c) Give a word description of the event $\{X\neq 0\ \text{and}\ Y\neq 0\}$ and compute the probability of this event.
- (d) Compute the marginal pmf of $X$ and of $Y$. Using $p_{X}(x)$, what is $P(X\leqslant 1)$?
- (e) Are $X$ and $Y$ independent rv's? Explain.