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# 问题

Questions in category: 概率论 (Probability).

## [Exer14-1] Exercise 1 of Book {Devore2017B} P.211

Posted by haifeng on 2020-05-27 18:31:28 last update 2020-05-27 18:42:01 | Answers (1) | 收藏

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.