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Questions in category: 概率论 (Probability).

## [Exer14-4] Exercise 7 of Book {Devore2017B} P.212

Posted by haifeng on 2020-05-27 18:47:08 last update 2020-05-27 18:47:08 | Answers (1) | 收藏

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.