问题及解答

The time $X$ (min) for a lab assistant to prepare the equipment for a certain experiment is believed to have a uniform distribution with $A=25$ and $B=35$.

• (a) Write the pdf of $X$ and sketch its graph.
• (b) What is the probability that preparation time exceeds $33$ min?
• (c) What is the probability that preparation time is within $2$ min of the mean time? [Hint: Identify $\mu$ from the graph of $f(x)$.]
• (d) For any $a$ such that $25 < a < a+2 < 35$, what is the probability that preparation time is between $a$ and $a+2$ min?
   

(a) The pdf is

\[
f(x)=\begin{cases}
\frac{1}{10}, & 25\leqslant x\leqslant 35,\\
0, & \text{otherwise}.
\end{cases}
\]

(b)

The probability that preparation time exceeds 33 min is

\[
P(X\geqslant 33)=\int_{33}^{35}f(x)dx=\int_{33}^{35}\frac{1}{10}dx=\frac{1}{10}\cdot(35-33)=\frac{1}{5}=0.2.
\]

(c)

First, the mean time of preparation time is equal to $\frac{25+35}{2}=30$.

The probability that preparation time is within 2 min of the mean time is

\[
P(30-2\leqslant X\leqslant 30+2)=\int_{28}^{32}f(x)dx=\int_{28}^{32}\frac{1}{10}dx=\frac{1}{10}\cdot(32-28)=0.4
\]

(d)

For any $a$ such that $25 < a < a+2 < 35$, the probability that preparation time is between $a$ and $a+2$ min is

\[
P(a\leqslant X\leqslant a+2)=\int_{a}^{a+2}f(x)dx=\int_{a}^{a+2}\frac{1}{10}dx=\frac{1}{10}\cdot(a-(a+2))=0.2
\]