[Exer14-2] Exercise 3 of Book {Devore2017B} P.212
A certain market has both an express checkout line and a superexpress checkout line. Let $X_1$ denote the number of customers in line at the express checkout at a particular time of day and let $X_2$ denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of $X_1$ and $X_2$ is as given in the accompanying table.
| $x_2$ | |||||
| 0 | 1 | 2 | 3 | ||
| 0 | .08 | .07 | .04 | .00 | |
| 1 | .06 | .15 | .05 | .04 | |
| $x_1$ | 2 | .05 | .04 | .10 | .06 |
| 3 | .00 | .03 | .04 | .07 | |
| 4 | .00 | .01 | .05 | .06 | |
%%Table in LaTeX
\begin{table}[htbp]
\centering
\begin{tabular}{cc|p{0.5in}p{0.5in}p{0.5in}p{0.5in}}
& & & $x_2$ & & \\
& & 0 & 1 & 2 & 3\\\hline
\multirow{5}{*}{$x_1$}& 0 & .08 & .07 & .04 & .00\\
~& 1 & .06 & .15 & .05 & .04\\
~& 2 & .05 & .04 & .10 & .06\\
~& 3 & .00 & .03 & .04 & .07\\
~& 4 & .00 & .01 & .05 & .06\\
\hline
\end{tabular}
\end{table}
- (a) What is $P(X_1=1,\ X_2=1)$, that is, the probability that there is exactly one customer in each line?
- (b) What is $P(X_1=X_2)$, that is, the probability that the numbers of customers in the two lines are identical?
- (c) Let $A$ denote the event that there are at least two more customers in one line than in the other line. Express $A$ in terms of $X_1$ and $X_2$, and calculate the probability of this event.
- (d) What is the probability that the total number of customers in the two lines is exactly four? At least four?