问题及解答

A mail-order computer business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table.

Calculate the probability of each of the following events.

• (a) $\{$ at most 3 lines are in use $\}$
• (b) $\{$ fewer than 3 lines are in use $\}$
• (c) $\{$ at least 3 lines are in use $\}$
• (d) $\{$ between 2 and 5 lines, inclusive, are in use $\}$
• (e) $\{$ between 2 and 4 lines, inclusive, are not in use $\}$
• (f) $\{$ at least 4 lines are not in use $\}$
   

Let $A,\,B,\,C,\,D,\,E,\,F$ stand for the events in the questions (a)--(f) respectively.

\[
P(A)=\sum_{x\leqslant 3}p(x)=p(0)+p(1)+p(2)+p(3)=0.10+0.15+0.20+0.25=0.70
\]

\[
P(B)=\sum_{x < 3}p(x)=p(0)+p(1)+p(2)=0.10+0.15+0.20=0.45
\]

\[
P(C)=\sum_{x\geqslant 3}p(x)=p(3)+p(4)+p(5)+p(6)=0.25+0.20+0.06+0.04=0.55
\]

\[
P(D)=\sum_{2\leqslant x\leqslant 5}p(x)=p(2)+p(3)+p(4)+p(5)=0.20+0.25+0.20+0.06=0.71
\]

\[
\begin{split}
P(E)&=P(\{\text{There are at most 1 line or at least 5 lines are in use}\})=\sum_{x\leqslant 1\ \text{or}\ x\geqslant 5}p(x)\\
&=p(0)+p(1)+p(5)+p(6)=0.10+0.15+0.06+0.04=0.35
\end{split}
\]

\[
\begin{split}
P(F)&=P(\{\text{at most 2 lines are in use}\})=\sum_{x\leqslant 2}p(x)\\
&=p(0)+p(1)+p(2)=0.10+0.15+0.20=0.45
\end{split}
\]