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Modern Mathematical Statistics with Applications

Devore, Jay L., Berk, Kenneth N.

Chapter 4

Continuous Random Variables and Probability Distributions - all with Video Answers

Educators

Section 1

Probability Density Functions and Cumulative Distribution Functions

03:49

Problem 1

- 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
$$
f(x)=\left\{\begin{array}{cl}
.5 x & 0 \leq x \leq 2 \\
0 & \text { otherwise }
\end{array}\right.
$$
Calculate the following probabilities:
a. $P(X \leq 1)$
b. $P(.5 \leq X \leq 1.5)$

BE
Blake Eriks
Numerade Educator
02:59

Problem 2

Suppose the reaction temperature $X\left(\right.$ in $\left.{ }^{\circ} \mathrm{C}\right)$ in a chemical process has a uniform distribution with $A=-5$ and $B=5$.
a. Compute $P(X<0)$.
b. Compute $P(-2.5<X<2.5)$.
c. Compute $P(-2 \leq X \leq 3)$.
d. For $k$ satisfying $-5<k<k+4<5$, compute $P(k<X<k+4)$. Interpret this in words.

Clarissa Noh
Clarissa Noh
Numerade Educator
14:23

Problem 3

Suppose the error involved in making a measurement is a continuous ry $X$ with pdf
$$
f(x)=\left\{\begin{array}{cc}
.09375\left(4-x^{2}\right) & -2 \leq x \leq 2 \\
0 & \text { otherwise }
\end{array}\right.
$$
a. Sketch the graph of $f(x)$.
b. Compute $P(X>0)$.
c. Compute $P(-1<X<1)$.
d. Compute $P(X<-.5$ or $X>.5)$.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
07:11

Problem 4

Let $X$ denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article "Blade Fatigue Life Assessment with Application to VAWTS" (J. Solar Energy Engrg., 1982: 107-111) proposes the Rayleigh distribution, with pdf
$$
f(x ; \theta)=\left\{\begin{array}{cl}
\frac{x}{\theta^{2}} \cdot e^{-x^{2} /\left(2 \theta^{2}\right)} & x>0 \\
0 & \text { otherwise }
\end{array}\right.
$$
as a model for the $X$ distribution.
a. Verify that $f(x ; \theta)$ is a legitimate pdf.
b. Suppose $\theta=100$ (a value suggested by a graph in the article). What is the probability that $X$ is at most 200 ? Less than 200 ? At least 200 ?
c. What is the probability that $X$ is between 100 and 200 (again assuming $\theta=100$ )?
d. Give an expression for $P(X \leq x)$.

Clarissa Noh
Clarissa Noh
Numerade Educator
09:49

Problem 5

A college professor never finishes his lecture before the end of the hour and always finishes his lectures within $2 \mathrm{~min}$ after the hour. Let $X=$ the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of $X$ is
$$
f(x)=\left\{\begin{array}{cc}
k x^{2} & 0 \leq x \leq 2 \\
0 & \text { otherwise }
\end{array}\right.
$$
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 $1 \mathrm{~min}$ of the end of the hour?
c. What is the probability that the lecture continues beyond the hour for between 60 and $90 \mathrm{~s}$ ?
d. What is the probability that the lecture continues for at least $90 \mathrm{~s}$ beyond the end of the hour?

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:48

Problem 6

The grade point averages (GPA's) for graduating seniors at a college are distributed as a continuous rv $X$ with pdf
$$
f(x)=\left\{\begin{array}{cc}
k\left[1-(x-3)^{2}\right] & 2 \leq x \leq 4 \\
0 & \text { otherwise }
\end{array}\right.
$$
a. Sketch the graph of $f(x)$.
b. Find the value of $k$.
c. Find the probability that a GPA exceeds $3 .$
d. Find the probability that a GPA is within $.25$ of 3 .
e. Find the probability that a GPA differs from 3 by more than $5 .$

James Kiss
James Kiss
Numerade Educator
03:34

Problem 7

The time $X(\mathrm{~min})$ 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 $33 \mathrm{~min}$ ?
c. What is the probability that preparation time is within $2 \mathrm{~min}$ 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 $a+2 \min$ ?

Michael Nartey
Michael Nartey
Numerade Educator
06:37

Problem 8

Commuting to work requires getting on a bus near home and then transferring to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with $A=0$ and $B=5$, then it can be shown that the total waiting time $Y$ has the pdf
$$
f(y)=\left\{\begin{array}{cl}
\frac{1}{25} y & 0 \leq y<5 \\
\frac{2}{5}-\frac{1}{25} y & 5 \leq y \leq 10 \\
0 & y<0 \text { or } y>10
\end{array}\right.
$$
a. Sketch the pdf of $Y$.
b. Verify that $\int_{-\infty}^{\infty} f(y) d y=1$.
c. What is the probability that total waiting time is at most 3 min?
d. What is the probability that total waiting time is at most $8 \mathrm{~min}$ ?
e. What is the probability that total waiting time is between 3 and 8 min?
f. What is the probability that total waiting time is either less than 2 min or more than 6 min?

Clarissa Noh
Clarissa Noh
Numerade Educator
05:06

Problem 9

Consider again the pdf of $X=$ time headway given in Example 4.5. What is the probability that time headway is
a. At most $6 \mathrm{~s}$ ?
b. More than $6 \mathrm{~s}$ ? At least $6 \mathrm{~s}$ ?
c. Between 5 and 6 s?

Robin Corrigan
Robin Corrigan
Numerade Educator
04:07

Problem 10

A family of pdf's that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, $k$ and $\theta$, both $>0$, and the pdf is
$$
f(x ; k, \theta)=\left\{\begin{array}{cc}
\frac{k \cdot \theta^{k}}{x^{k+1}} & x \geq \theta \\
0 & x<\theta
\end{array}\right.
$$
a. Sketch the graph of $f(x ; k, \theta)$.
b. Verify that the total area under the graph equals 1 .
c. If the rv $X$ has pdf $f(x ; k, \theta)$, for any fixed $b>\theta$, obtain an expression for $P(X \leq b)$.
d. For $\theta<a<b$, obtain an expression for the probability $P(a \leq X \leq b)$.

Clarissa Noh
Clarissa Noh
Numerade Educator
03:49

Problem 11

The cdf of checkout duration $X$ as described in Exercise 1 is
$$
F(x)= \begin{cases}0 & x<0 \\ \frac{x^{2}}{4} & 0 \leq x<2 \\ 1 & 2 \leq x\end{cases}
$$Use this to compute the following:
a. $P(X \leq 1)$
b. $P(.5 \leq X \leq 1)$
c. $P(X>.5)$
d. The median checkout duration $\bar{\mu}[$ solve $.5=$ $F(\bar{\mu})]$
e. $F^{\prime}(x)$ to obtain the density function $f(x)$

BE
Blake Eriks
Numerade Educator
05:26

Problem 12

The cdf for $X$ ( $=$ measurement error) of Exercise 3 is
$$
F(x)=\left\{\begin{array}{cl}
0 & x<-2 \\
\frac{1}{2}+\frac{3}{32}\left(4 x-\frac{x^{3}}{3}\right) & -2 \leq x<2 \\
1 & 2 \leq x
\end{array}\right.
$$
a. Compute $P(X<0)$.
b. Compute $P(-1<X<1)$.
c. Compute $P(.5<X)$.
d. Verify that $f(x)$ is as given in Exercise 3 by obtaining $F^{\prime}(x)$.
e. Verify that $\bar{\mu}=0$.

Clarissa Noh
Clarissa Noh
Numerade Educator
05:45

Problem 13

Example $4.5$ introduced the concept of time headway in traffic flow and proposed a particular distribution for $X=$ the headway between two randomly selected consecutive cars (sec). Suppose that in a different traffic environment, the distribution of time headway has the form
$$
f(x)= \begin{cases}\frac{k}{x^{4}} & x>1 \\ 0 & x \leq 1\end{cases}
$$
a. Determine the value of $k$ for which $f(x)$ is a legitimate pdf.
b. Obtain the cumulative distribution function.
c. Use the cdf from (b) to determine the probability that headway exceeds $2 \mathrm{~s}$ and also the probability that headway is between 2 and $3 \mathrm{~s}$.

Robin Corrigan
Robin Corrigan
Numerade Educator
05:25

Problem 14

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
$$
f(x)=\left\{\begin{array}{cl}
90 x^{8}(1-x) & 0<x<1 \\
0 & \text { otherwise }
\end{array}\right.
$$
a. Graph the pdf. Then obtain the cdf of $X$ and graph it.
b. What is $P(X \leq .5)[$ i.e., $F(.5)]$ ?
c. Using part (a), what is $P(.25<X \leq .5)$ ? What is $P(.25 \leq X \leq .5)$ ?
d. What is the 75 th percentile of the distribution?

Clarissa Noh
Clarissa Noh
Numerade Educator
06:42

Problem 15

Answer parts (a)-(d) of Exercise 14 for the random variable $X$, lecture time past the hour, given in Exercise $5 .$

Robin Corrigan
Robin Corrigan
Numerade Educator
10:27

Problem 16

Let $X$ be a continuous rv with cdf
$$
F(x)=\left\{\begin{array}{cl}
0 & x \leq 0 \\
\frac{x}{4}\left[1+\ln \left(\frac{4}{x}\right)\right] & 0<x \leq 4 \\
1 & x>4
\end{array}\right.
$$
[This type of cdf is suggested in the article "Variability in Measured Bedload-Transport Rates" (Water Resources Bull., 1985:39-48) as a model for a hydrologic variable.] What is
a. $P(X \leq 1)$ ?
b. $P(1 \leq X \leq 3)$ ?
c. The pdf of $X$ ?

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
04:13

Problem 17

Let $X$ be the temperature in ${ }^{\circ} \mathrm{C}$ at which a chemical reaction takes place, and let $Y$ be the temperature in ${ }^{\circ} \mathrm{F}$ (so $Y=1.8 X+32$ ).
a. If the median of the $X$ distribution is $\bar{\mu}$, show that $1.8 \bar{\mu}+32$ is the median of the $Y$ distribution.
b. How is the 90 th percentile of the $Y$ distribution related to the 90 th percentile of the $X$ distribution? Verify your conjecture.
c. More generally, if $Y=a X+b$, how is any particular percentile of the $Y$ distribution related to the corresponding percentile of the $X$ distribution?

Clarissa Noh
Clarissa Noh
Numerade Educator