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Finite Mathematics and Calculus with Applications

Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Chapter 18

Probability and Calculus

Educators


Problem 1

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{1}{9} x-\frac{1}{18} ;[2,5]$$

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Problem 2

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{1}{3} x-\frac{1}{6} ;[3,4]$$

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Problem 3

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{x^{2}}{21} ;[1,4]$$

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Problem 4

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{3}{98} x^{2} ;[3,5]$$

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Problem 5

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=4 x^{3} ;[0,3]$$

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Problem 6

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{x^{3}}{81} ;[0,3]$$

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Problem 7

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{x^{2}}{16} ;[-2,2]$$

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Problem 8

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=2 x^{2} ;[-1,1]$$

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Problem 9

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{5}{3} x^{2}-\frac{5}{90} ;[-1,1]$$

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Problem 10

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why.
$$f(x)=\frac{3}{13} x^{2}-\frac{12}{13} x+\frac{45}{52} ;[0,4]$$

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Problem 11

Find a value of $k$ that will make $f$ a probability density function on the indicated interval.
$$f(x)=k x^{1 / 2} ;[1,4]$$

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Problem 12

Find a value of $k$ that will make $f$ a probability density function on the indicated interval.
$$f(x)=k x^{3 / 2} ;[4,9]$$

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Problem 13

Find a value of $k$ that will make $f$ a probability density function on the indicated interval.
$$f(x)=k x^{2} ;[0,5]$$

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Problem 14

Find a value of $k$ that will make $f$ a probability density function on the indicated interval.
$$f(x)=k x^{2} ;[-1,2]$$

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Problem 15

Find a value of $k$ that will make $f$ a probability density function on the indicated interval.
$$f(x)=k x ;[0,3]$$

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Problem 16

Find a value of $k$ that will make $f$ a probability density function on the indicated interval.
$$f(x)=k x ;[2,3]$$

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Problem 17

Find a value of $k$ that will make $f$ a probability density function on the indicated interval.
$$f(x)=k x ;[1,5]$$

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Problem 18

Find a value of $k$ that will make $f$ a probability density function on the indicated interval.
$$f(x)=k x^{3} ;[2,4]$$

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Problem 19

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 1

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Problem 20

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 2

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Problem 21

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 3

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Problem 22

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 4

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Problem 23

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 11

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Problem 24

Find the cumulative distribution function for the probability density function in each of the following exercises.
Exercise 12

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Problem 25

The total area under the graph of a probability density function always equals __________.

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Problem 26

In your own words, define a random variable.

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Problem 27

What is the difference between a discrete probability function and a probability density function?

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Problem 28

Why is $P(X=c)=0$ for any number $c$ in the domain of a probability density function?

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Problem 29

Show that each function defined as follows is a probability density function on the given interval; then find the indicated probabilities.
$$f(x)=\frac{1}{2}(1+x)^{-3 / 2} ;[0, \infty)$$
a. $P(0 \leq X \leq 2) \qquad$ b. $P(1 \leq X \leq 3)$
c. $P(X \geq 5)$

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Problem 30

Show that each function defined as follows is a probability density function on the given interval; then find the indicated probabilities.
$$f(x)=e^{-x} ;[0, \infty)$$
a. $P(0 \leq X \leq 1) \qquad$ b. $P(1 \leq X \leq 2)$
c. $P(X \leq 2)$

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Problem 31

Show that each function defined as follows is a probability density function on the given interval; then find the indicated probabilities.
$$f(x)=(1 / 2) e^{-x / 2} ;[0, \infty)$$
a. $P(0 \leq X \leq 1) \qquad$ b. $P(1 \leq X \leq 3)$
c. $P(X \geq 2)$

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Problem 32

Show that each function defined as follows is a probability density function on the given interval; then find the indicated probabilities.
$$f(x)=\frac{20}{(x+20)^{2}} ;[0, \infty)$$
a. $P(0 \leq X \leq 1) \qquad$ b. $P(1 \leq X \leq 5)$
c. $P(X \geq 5)$

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Problem 33

Show that each function defined as follows is a probability density function on the given interval; then find the indicated probabilities.
$$f(x)=\left\{\begin{array}{ll}{\frac{x^{3}}{12}} & {\text { if } 0 \leq x \leq 2} \\ {\frac{16}{3 x^{3}}} & {\text { if } x>2}\end{array}\right.$$
a. $P(0 \leq X \leq 2) \qquad$ b. $P(X \geq 2)$
c. $P(1 \leq X \leq 3)$

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Problem 34

Show that each function defined as follows is a probability density function on the given interval; then find the indicated probabilities.
$$f(x)=\left\{\begin{array}{ll}{\frac{20 x^{4}}{9}} & {\text { if } 0 \leq x \leq 1} \\ {\frac{20}{9 x^{5}}} & {\text { if } x>1}\end{array}\right.$$
a. $P(0 \leq X \leq 1) \qquad$ b. $P(X \geq 1)$
c. $P(0 \leq X \leq 2)$

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Problem 35

Life Span of a Computer Part The life (in months) of a certain electronic computer part has a probability density function defined by
$$f(t)=\frac{1}{2} e^{-t / 2} \text { for } t \text { in }[0, \infty) \text { . }$$
Find the probability that a randomly selected component will last the following lengths of time.
a. At most 12 months
b. Between 12 and 20 months
c. Find the cumulative distribution function for this random variable.
d. Use the answer to part c to find the probability that a randomly selected component will last at most 6 months.

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Problem 36

Machine Life A machine has a useful life of 4 to 9 years, and its life (in years) has a probability density function defined by
$$f(t)=\frac{1}{11}\left(1+\frac{3}{\sqrt{t}}\right)$$
Find the probabilities that the useful life of such a machine selected at random will be the following.
a. Longer than 6 years
b. Less than 5 years
c. Between 4 and 7 years
d. Find the cumulative distribution function for this random variable.
e. Use the answer to part d to find the probability that a randomly selected machine has a useful life of at most 8 years.

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Problem 37

Machine Part The lifetime of a machine part has a continuous distribution on the interval $(0,40)$ with probability density function $f,$ where $f(x)$ is proportional to $(10+x)^{-2}$ . Calculate the probability that the lifetime of the machine part is less than $6 .$ Choose one of the following. Source: Society of Actuaries.
a. $0.04 \qquad$ b. $0.15 \qquad$ c. $0.47 \qquad$ d. $0.53 \qquad$ e. $0.94$

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Problem 38

Insurance An insurance policy pays for a random loss $X$ subject to a deductible of $C,$ where $0 < C < 1 .$ The loss amount is modeled as a continuous random variable with density function
$$f(x)=\left\{\begin{array}{ll}{2 x} & {\text { for } 0 < x < 1} \\ {0} & {\text { otherwise }}\end{array}\right.$$
Given a random loss $X,$ the probability that the insurance payment is less than 0.5 is equal to 0.64 . Calculate $C$ . Choose one of the following. (Hint: The payment is 0 unless the loss is greater than the deductible, in which case the payment is the loss minus the deductible.) Source: Society of Actuaries.
a. $0.1 \qquad$ b. $0.3 \qquad$ c. $0.4 \qquad$ d. $0.6 \qquad$ e. $0.8$

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Problem 39

Petal Length The length of a petal on a certain flower varies from 1 $\mathrm{cm}$ to 4 $\mathrm{cm}$ and has a probability density function defined by
$$f(x)=\frac{1}{2 \sqrt{x}}$$
Find the probabilities that the length of a randomly selected petal will be as follows.
a. Greater than or equal to 3 cm
b. Less than or equal to 2 cm
c. Between 2 cm and 3 cm

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Problem 40

Clotting Time of Blood The clotting time of blood is a random variable $t$ with values from 1 second to 20 seconds and probability density function defined by
$$f(t)=\frac{1}{(\ln 20) t}.$$
Find the following probabilities for a person selected at random.
a. The probability that the clotting time is between 1 and 5 seconds
b. The probability that the clotting time is greater than 10 seconds

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Problem 41

Flour Beetles Researchers who study the abundance of the flour beetle, Tribolium castaneum, have developed a probability density function that can be used to estimate the abundance of the beetle in a population. The density function, which is a member of the gamma distribution, is
$$f(x)=1.185 \times 10^{-9} x^{4.52222} e^{-0.049846 x},$$
where $x$ is the size of the population. Source: Ecology.
a. Estimate the probability that a randomly selected flour beetle population is between 0 and 150.
b. Estimate the probability that a randomly selected flour beetle population is between 100 and 200.

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Problem 42

Flea Beetles The mobility of an insect is an important part of its survival. Researchers have determined that the probability that a marked flea beetle, Phyllotreta cruciferae and Phyllotreta striolata, will be recaptured within a certain distance and time after release can be calculated from the probability density function
$$p(x, t)=\frac{e^{-x^{2} /(4 D t)}}{\int_{0}^{L} e^{-u^{2} /(4 D t)} d u},$$
where $t$ is the time after release (in hours), $x$ is the distance (in meters) from the release point that recaptures occur, $L$ is the maximum distance from the release point that recaptures can occur, and $D$ is the diffusion coefficient. Source: Ecology Monographs.
a. If $t=12, L=6,$ and $D=38.3$ , find the probability that a flea
b. Using the same values for $t, L,$ and $D,$ find the probability that a flea beetle will be recaptured between 1 and 5 $\mathrm{m}$ of the release point.

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Problem 43

Social Network The number of U.S. users (in millions) on Facebook, a computer social network, in 2009 is given in the table below. Source: Inside Facebook.
$$\begin{array}{|c|c|}\hline {} & {} & \text { Number of Users } \\ \text { Age Interval } & {\text { Midpoint of }} & {\text { in Each Interval }} \\ \text { (years) } & {\text { Interval (year) }} & {\text { (millions) }} \\ \hline 13-17 & {15} & {6.049} \\ {18-25} & {21.5} & {19.461} \\ {26-34} & {30} & {13.423} \\ {35-44} & {39.5} & {9.701} \\ {45-54} & {49.5} & {4.582} \\ {55-65} & {60} & {2.849} \\ {\text { Total }} & {} & {56.065} \\ \hline \end{array}$$
a. Plot the data. What type of function appears to best match this data?
b. Use the regression feature on your graphing calculator to find a quartic equation that models the number of years, $t,$ since birth and the number of Facebook users, $N(t)$ Use the midpoint value to estimate the point in each interval for the age of the Facebook user. Graph the function with the plot of the data. Does the function resemble the data?
c. By finding an appropriate constant $k,$ find a function $S(t)=k N(t)$ that is a probability density function describing the probability of the age of a Facebook user. (Hint: Because the function in part b is negative for values less than 13.4 and greater than $62.0,$ restrict the domain of the density function to the interval $[13.4,62.01 . \text { That is, }$ integrate the function you found in part b from 13.4 to $62.0 .$ )
d. For a randomly chosen person who uses Facebook, find the probabilities that the person was at least 35 but less than 45 years old, at least 18 but less than 35 years old, and at least 45 years old. Compare these with the actual probabilities.

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Problem 44

Time to Learn a Task The time required for a person to learn a certain task is a random variable with probability density function defined by
$$f(t)=\frac{8}{7(t-2)^{2}}.$$
The time required to learn the task is between 3 and 10 minutes. Find the probabilities that a randomly selected person will learn the task in the following lengths of time.
a. Less than 4 minutes
b. More than 5 minutes

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Problem 45

Annual Rainfall The annual rainfall in a remote Middle Eastern country varies from 0 to 5 in. and is a random variable with probability density function defined by
$$f(x)=\frac{5.5-x}{15}.$$
Find the following probabilities for the annual rainfall in a randomly selected year.
a. The probability that the annual rainfall is greater than 3 in.
b. The probability that the annual rainfall is less than 2 in.
c. The probability that the annual rainfall is between 1 in. and 4 in.

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Problem 46

Earthquakes The time between major earthquakes in the Southern California region is a random variable with probability density function
$$f(t)=\frac{1}{960} e^{-t / 960},$$
where $t$ is measured in days. Source: Journal of Seismology.
a. Find the probability that the time between a major earthquake and the next one is less than 365 days.
b. Find the probability that the time between a major earthquake and the next one is more than 960 days.

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Problem 47

Earthquakes The time between major earthquakes in the Taiwan region is a random variable with probability density function
$$f(t)=\frac{1}{3650.1} e^{-t / 3650.1} $$
where $t$ is measured in days. Source: Journal of Seismology.
a. Find the probability that the time between a major earthquake and the next one is more than 1 year but less than 3 years.
b. Find the probability that the time between a major earthquake and the next one is more than 7300 days.

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Problem 48

Drunk Drivers The frequency of alcohol-related traffic fatalities has dropped in recent years but is still high among young people. Based on data from the National Highway Traffic Safety Administration, the age of a randomly selected, alcohol-impaired driver in a fatal car crash is a random variable with probability density function given by
$$f(t)=\frac{4.045}{t^{1.532}} \text { for } t \text { in }[16,80].$$
Find the following probabilities of the age of such a driver. Source: Traffic Safety Facts.
a. Less than or equal to 25
b. Greater than or equal to 35
c. Between 21 and 30
d. Find the cumulative distribution function for this random variable.
e. Use the answer to part d to find the probability that a randomly selected alcohol-impaired driver in a fatal car crash is at most 21 years old.

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Problem 49

Driving Fatalities We saw in a review exercise in Chapter 12 on Calculating the Derivative that driver fatality rates were highest for the youngest and oldest drivers. When adjusted for the number of miles driven by people in each age group, the number of drivers in fatal crashes goes down with age, and the age of a randomly selected driver in a fatal car crash is a random variable with probability density function given by
$$f(t)=0.06049 e^{-0.03211 t} \quad \text { for } t \text { in }[16,84]$$
Find the following probabilities of the age of such a driver. Source: National Highway Traffic Safety Administration.
a. Less than or equal to 25
b. Greater than or equal to 35
c. Between 21 and 30
d. Find the cumulative distribution function for this random variable.
e. Use the answer to part d to find the probability that a randomly selected driver in a fatal crash is at most 21 years old.

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Problem 50

Length of a Telephone Call The length of a telephone call minutes), $t,$ for a certain town is a continuous random variable with probability density function defined by
$$f(t)=3 t^{-4}, \text { for } t \text { in }[1, \infty).$$
Find the probabilities for the following situations.
a. The call lasts between 1 and 2 minutes.
b. The call lasts between 3 and 5 minutes.
c. The call lasts longer than 3 minutes.

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Problem 51

Time of Traffic Fatality The National Highway Traffic Safety Administration records the time of day of fatal crashes. The following table gives the time of day (in hours since midnight) and the frequency of fatal crashes. Source: The National Highway Traffic Safety Administration.
$$\begin{array}{|c|c|}\hline {} & \text { Midpoint of } & {}\\ \text { Time of Day } & {\text { Interval (hours) }} & {\text { Frequency }} \\ \hline 0-3 & {1.5} & {4486} \\ \hline 3-6 & {4.5} & {2774} \\ \hline 6-9 & {7.5} & {3236} \\ \hline 9-12 & {10.5} & {3285} \\ \hline 12-15 & {13.5} & {4356} \\ \hline {15-18} & {16.5} & {5325} \\ \hline {18-21} & \hline {19.5} & {5342} \\ \hline {21-24} & {22.5} & {4952} \\ \hline {\text { Total }} & {} & {33,756} \\ \hline\end{array}$$
a. Plot the data. What type of function appears to best match this data?
b. Use the regression feature on your graphing calculator to find a cubic equation that models the time of day, $t,$ and the number of traffic fatalities, $T(t) .$ Use the midpoint value to estimate the time in each interval. Graph the function with the plot of the data. Does the graph fit the data?
c. By finding an appropriate constant $k,$ find a function $S(t)=$ $k T(t)$ that is a probability density function describing the probability of a traffic fatality at a particular time of day.
d. For a randomly chosen traffic fatality, find the probabilities that the accident occurred between 12 am and 2 am $(t=0$ to $t=2 )$ and between 4 pm and $5 : 30$ pm $(t=16 \text { to }$ $t=17.5$ )

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