Let $X$ denote the amount of space occupied by an article placed in a $1$-$\mathrm{ft}^3$ packing container. The pdf of $X$ is
\[
f(x)=
\begin{cases}
  90x^8(1-x), & 0 < x < 1, \\
  0, & \mbox{otherwise}.
\end{cases}
\]

• (a) Graph the pdf. Then obtain the cdf of $X$ and graph it.
• (b) What is $P(X\leqslant .5)$ [i.e., $F(.5)$]?
• (c) Using part (a), what is $P(.25 < X \leqslant .5)$? What is $P(.25\leqslant X\leqslant .5)$?
• (d) What is the $75$th percentile of the distribution?
• (e) Compute $E(X)$ and $\sigma_X$.
• (f) What is the probability that $X$ is within $1$ standard deviation of its mean value?
   

Let $X$ denote checkout time duration with pdf given in Exercise 1 , 

\[
f(x)=\begin{cases}
       .5x, & 0\leqslant x\leqslant 2, \\
       0, & \mbox{otherwise}.
     \end{cases}
\]

• (a) Compute $E(X)$.
• (b) Compute $V(X)$ and $\sigma_X$.
• (c) If the borrower is charged an amount $h(X)=X^2$ when checkout duration is $X$, compute the expected charge $E[h(X)]$.
   

The cdf of checkout duration $X$ as described in Exercise 1 (见 Exer9-1) is
\[
F(x)=\begin{cases}
       0, & x < 0, \\
       \frac{x^2}{4}, & 0\leqslant x < 2,\\
       1, & 2\leqslant x.
     \end{cases}
\]
Use this to compute the following:

• (a) $P(X\leqslant 1)$;
• (b) $P(.5\leqslant X\leqslant 1.5)$ and $P(.5\leqslant X\leqslant 1)$.
• (c) $P(X > .5)$;
• (d) The median checkout duration $\tilde{\mu}$ [solve $.5=F(\tilde{\mu})$];
• (e) $F'(x)$ to obtain the density function $f(x)$.
   

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 college professor never finishes his lecture before the bell rings to end the period and always finishes his lectures within $2$ min after the bell rings. Let $X$= the time that elapses between the bell and the end of the lecture and suppose the pdf of $X$ is
\[
f(x)=
\begin{cases}
  kx^2, & 0\leqslant x\leqslant 2, \\
  0, & \mbox{otherwise}.
\end{cases}
\]

• (a) Find the value of $k$. [Hint: Total area under the graph of $f(x)$ is $1$.]
• (b) What is the probability that the lecture ends within $1$ min of the bell ringing?
• (c) What is the probability that the lecture continues beyond the bell for between $60$ and $90$ sec?
• (d) What is the probability that the lecture continues for at least $90$ sec beyond the bell?
   

Suppose the error involved in making a certain measurement is a continuous rv $X$ with pdf
\[
f(x)=
\begin{cases}
  .09375(4-x^2), & -2\leqslant x\leqslant 2, \\
  0, & \mbox{otherwise}.
\end{cases}
\]

• (a) Sketch the graph of $f(x)$.
• (b) Compute $P(X>0)$.
• (c) Compute $P(-1<X<1)$.
• (d) Compute $P(X<-.5\ \text{or}\ X>.5)$.
   

Let $X$ denote the amount of time for which a book on $2$-hour reserve at a college library is checked out by a randomly selected student and suppose that $X$ has density function
\[
f(x)=\begin{cases}
       .5x, & 0\leqslant x\leqslant 2, \\
       0, & \mbox{otherwise}.
     \end{cases}
\]
Calculate the following probabilities:

• (a) $P(X\leqslant 1)$
• (b) $P(.5\leqslant X\leqslant 1.5)$
• (c) $P(1.5 < X )$ 

[正體中文]

令 $X$ 表示由隨機選擇的學生簽出大學圖書館 $2$-小時儲備的書籍的時間量, 並假設 $X$ 具有密度函數

\[
f(x)=\begin{cases}
       .5x, & 0\leqslant x\leqslant 2, \\
       0, & \mbox{其他}.
     \end{cases}
\]
計算以下概率:

• (a) $P(X\leqslant 1)$
• (b) $P(.5\leqslant X\leqslant 1.5)$
• (c) $P(1.5 < X )$ 

 [Français]

Laissez $X$ indiquer le temps pour lequel un livre sur une réserve de $2$ heures dans une bibliothèque collégiale est vérifié par un étudiant choisi au hasard et supposons que $X$ a la fonction de densité

\[
f(x)=\begin{cases}
       .5x, & 0\leqslant x\leqslant 2, \\
       0, & \mbox{sinon}.
     \end{cases}
\]
Calculer les probabilités suivantes :

• (a) $P(X\leqslant 1)$
• (b) $P(.5\leqslant X\leqslant 1.5)$
• (c) $P(1.5 < X )$ 

泊松过程(Poisson process)

A very important application of the Poisson distribution arises in connection with the occurrence of events of a particular type over time. As an example, suppose that starting from a time point that we label $t=0$, we are interested in counting the number of radioactive pulses(放射性脉冲) recorded by a Geiger counter(盖革计数器). We make the following assumptions about the way in which pulses occur:

• 1. There exists a parameter $\alpha > 0$ such that for any short time interval of length $\Delta t$, the probability that exactly one pulse is receive is $\alpha\cdot\Delta t+o(\Delta t)$.
• 2. The probability of more than one pulse being received during $\Delta t$ is $o(\Delta t)$ [which, along with Assumption 1, implies that the probability of no pulses during $\Delta t$ is $1-\alpha\cdot\Delta t-o(\Delta t)$].
• 3. The number of pulses received during the time interval $\Delta t$ is independent of the number received prior to this time interval.
   

Informally, Assumption 1 says that, for a short interval of time, the probability of receiving a single pulse is approximately proportional to the length of the time interval, where $\alpha$ is the constant of proportionality.

Now let $P_k(t)$ denote the probability that $k$ pulses will be received by the counter during any particular time interval of length $t$.

Proposition. $P_k(t)=\frac{e^{-\alpha t}\cdot(\alpha t)}{k!}$, so that the number of pulses during a time interval of length $t$ is a Poisson rv with parameter $\lambda=\alpha t$. The expected number of pulses during any such time interval is then $\alpha t$, so the expected number during a unit interval of time is $\alpha$.
 

References:

The above content in English is copied from the following book:

《Probability and Statistics For Engineering and The Sciences》(Fifth Edtion) P.131
Author: Jay L. Devore

Section 5 of Chapter 3.

The number of requests for assistance received by a towing service(牵引服务/牽引服務/service de remorquage) is a Poisson process with rate $\alpha=4$ per hour.

• (a) Compute the probability that exactly ten requests are received during a particular $2$-hour period.
• (b) If the operators of the towing services take a $30$-min break for lunch, what is the probability that they do not miss any calls for assistance?
• (c) How many calls would you expect during their break?
   

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate $\alpha=8$ per hour, so that the number of arrivals during a time period of $t$ hours is a Poisson rv with parameter $\lambda=8t$.

• (a) What is the probability that exactly $5$ small aircraft arrive during a $1$-hour period? At least $5$? At least 10?
• (b) What are the expected value and standard deviation of the number of small aircraft that arrive during a $90$-min period?
• (c) What is the probability that at least $20$ small aircraft arrive during a $2\frac{1}{2}$-hour period? That at most $10$ arrive during this period?