Calculus Single Variable

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 8

Using the Definite Integral - all with Video Answers

Educators

Section 8

Probability, Mean, and Median

Problem 1

Show that the area under the fishing density function in Figure 8.103 on page 464 is $1 .$ Why is this to be expected?

Ahmad Reda

Ahmad Reda

Numerade Educator

Problem 2

Find the mean daily catch for the fishing data in Figure $8.103,$ page 464.

Ahmad Reda

Numerade Educator

Problem 3

(a) Using a calculator or computer, sketch graphs of the density function of the normal distribution $p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} /\left(2 \sigma^{2}\right)}$.
(i) For fixed $\mu$ (say, $\mu=5$ ) and varying $\sigma$ (say, $\sigma=1,2,3)$
(ii) For varying $\mu$ (say, $\mu=4,5,6$ ) and fixed $\sigma$ $(\text { say, } \sigma=1)$
(b) Explain how the graphs confirm that $\mu$ is the mean of the distribution and that $\sigma$ is a measure of how closely the data is clustered around the mean.

Ahmad Reda

Numerade Educator

Problem 4

A density function $p(x)$ satisfying $(\mathrm{I})-(\mathrm{IV})$ gives the fraction of years with a given total annual snowfall (in $\mathrm{m}$ ) for a city.
I. $\int_{0}^{0.5} p(x) d x=0.1$
II. $\int_{0}^{2} p(x) d x=0.3$
III. $\int_{0}^{2.72} p(x) d x=0.5$
IV. $\int_{0}^{\infty} x p(x) d x=2.65$
What is the median annual snowfall (in $\mathrm{m}$ )?

Ahmad Reda

Numerade Educator

Problem 5

A density function $p(x)$ satisfying $(\mathrm{I})-(\mathrm{IV})$ gives the fraction of years with a given total annual snowfall (in $\mathrm{m}$ ) for a city.
I. $\int_{0}^{0.5} p(x) d x=0.1$
II. $\int_{0}^{2} p(x) d x=0.3$
III. $\int_{0}^{2.72} p(x) d x=0.5$
IV. $\int_{0}^{\infty} x p(x) d x=2.65$
What is the mean snowfall (in $\mathrm{m}$ )?

Ahmad Reda

Numerade Educator

Problem 6

A density function $p(x)$ satisfying $(\mathrm{I})-(\mathrm{IV})$ gives the fraction of years with a given total annual snowfall (in $\mathrm{m}$ ) for a city.
I. $\int_{0}^{0.5} p(x) d x=0.1$
II. $\int_{0}^{2} p(x) d x=0.3$
III. $\int_{0}^{2.72} p(x) d x=0.5$
IV. $\int_{0}^{\infty} x p(x) d x=2.65$
What is the probability of an annual snowfall between 0.5 and $2 \mathrm{m} ?$

Ahmad Reda

Numerade Educator

Problem 7

A screening test for susceptibility to diabetes reports a numerical score between 0 to $100 .$ A score greater than 50 indicates a potential risk, with some lifestyle training recommended. Results from 200,000 people who were tested show that :
$\bullet$ $75 \%$ received scores evenly distributed between 0 and $50 .$
$\bullet$ $25 \%$ received scores evenly distributed between 50 and 100
The probability density function (pdf) is in Figure 8.110
(a) Find the values of $A$ and $B$ that make this a probability density function.
(b) Find the median test score.
(c) Find the mean test score.
(d) Give a graph of the cumulative distribution function (cdf) for these test scores.

Ahmad Reda

Numerade Educator

Problem 8

Use Figure $8.111,$ a graph of $p(x),$ a density function for the fraction of a region's winters with a given total snowfall (in $\mathrm{m}$ ).
Which of the following events is most likely?
I. A winter has $3 \mathrm{m}$ or more of snow?
II. A winter has $2 \mathrm{m}$ or less of snow?
III. A winter with between 2 and $3 \mathrm{m}$ of snow?

Ahmad Reda

Numerade Educator

Problem 9

Use Figure $8.111,$ a graph of $p(x),$ a density function for the fraction of a region's winters with a given total snowfall (in $\mathrm{m}$ ).
The shaded area is 0.8. What percentage of winters see more than 3 meters of total snowfall?

Ahmad Reda

Numerade Educator

Problem 10

Use Figure $8.111,$ a graph of $p(x),$ a density function for the fraction of a region's winters with a given total snowfall (in $\mathrm{m}$ ).
What appears to be the smallest and largest total annual snowfall?

Ahmad Reda

Numerade Educator

Problem 11

Use Figure $8.111,$ a graph of $p(x),$ a density function for the fraction of a region's winters with a given total snowfall (in $\mathrm{m}$ ).
If $p(2.8)=1.1,$ approximately what percentage of winters see snowfall totals between $2.8 \mathrm{m}$ and $3.0 \mathrm{m} ?$

Ahmad Reda

Numerade Educator

Problem 11

Use Figure $8.111,$ a graph of $p(x),$ a density function for the fraction of a region's winters with a given total snowfall (in $\mathrm{m}$ ).
If $p(2.8)=1.1,$ approximately what percentage of winters see snowfall totals between $2.8 \mathrm{m}$ and $3.0 \mathrm{m} ?$

Ahmad Reda

Numerade Educator

Problem 12

A quantity $x$ has density function $p(x)=0.5(2-x)$ for $0 \leq x \leq 2$ and $p(x)=0$ otherwise. Find the mean and median of $x$.

Ahmad Reda

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Problem 13

A quantity $x$ has cumulative distribution function $P(x)=x-x^{2} / 4$ for $0 \leq x \leq 2$ and $P(x)=0$ for $x<0$ and $P(x)=1$ for $x>2 .$ Find the mean and median of $x$.

Ahmad Reda

Numerade Educator

Problem 14

The probability of a transistor failing between $t=a$ months and $t=b$ months is given by $c \int_{a}^{b} e^{-c t} d t,$ for some constant $c$.
(a) If the probability of failure within the first six months is $10 \%,$ what is $c ?$
(b) Given the value of $c$ in part (a), what is the probability the transistor fails within the second six months?

Ahmad Reda

Numerade Educator

Problem 15

Suppose that $x$ measures the time (in hours) it takes for a student to complete an exam. All students are done within two hours and the density function for $x$ is $p(x)=\left\{\begin{array}{ll}x^{3} / 4 & \text { if } 0<x<2 \\ 0 & \text { otherwise }\end{array}\right.$
(a) What proportion of students take between 1.5 and 2.0 hours to finish the exam?
(b) What is the mean time for students to complete the exam?
(c) Compute the median of this distribution.

Ahmad Reda

Numerade Educator

Problem 16

In 1950 an experiment was done observing the time gaps between successive cars on the Arroyo Seco Freeway. ${ }^{10}$ The data show that the density function of these time gaps was given approximately by $p(x)=a e^{-0.122 x}$ where $x$ is the time in seconds and $a$ is a constant.
(a) Find $a$
(b) Find $P$, the cumulative distribution function.
(c) Find the median and mean time gap.
(d) Sketch rough graphs of $p$ and $P$.

Ahmad Reda

Numerade Educator

Problem 17

Consider a group of people who have received treatment for a disease such as cancer. Let $t$ be the survival time, the number of years a person lives after receiving treatment. The density function giving the distribution of $t$ is $p(t)=C e^{-C t}$ for some positive constant $C$.
(a) What is the practical meaning for the cumulative distribution function $P(t)=\int_{0}^{t} p(x) d x ?$
(b) The survival function, $S(t),$ is the probability that a randomly selected person survives for at least $t$ years. Find $S(t)$.
(c) Suppose a patient has a $70 \%$ probability of surviving at least two years. Find $C$.

Ahmad Reda

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Problem 18

While taking a walk along the road where you live, you accidentally drop your glove, but you don't know where. The probability density $p(x)$ for having dropped the glove $x$ kilometers from home (along the road) is $p(x)=2 e^{-2 x} \quad$ for $x \geq 0$.
(a) What is the probability that you dropped it within 1 kilometer of home?
(b) At what distance $y$ from home is the probability that you dropped it within $y \mathrm{km}$ of home equal to $0.95 ?$

Ahmad Reda

Numerade Educator

Problem 19

The distribution of IQ scores can be modeled by a normal distribution with mean 100 and standard deviation
15.
(a) Write the formula for the density function of IQ scores.
(b) Estimate the fraction of the population with IQ between 115 and 120 .

Ahmad Reda

Numerade Educator

Problem 20

The speeds of cars on a road are approximately normally distributed with a mean $\mu=58 \mathrm{km} / \mathrm{hr}$ and standard deviation $\sigma=4 \mathrm{km} / \mathrm{hr}$.
(a) What is the probability that a randomly selected car is going between 60 and $65 \mathrm{km} / \mathrm{hr}$ ?
(b) What fraction of all cars are going slower than 52 $\mathrm{km} / \mathrm{hr} ?$

Ahmad Reda

Numerade Educator

Problem 21

Consider the normal distribution, $p(x)$.
(a) Show that $p(x)$ is a maximum when $x=\mu .$ What is that maximum value?
(b) Show that $p(x)$ has points of inflection where $x=$ $\mu+\sigma$ and $x=\mu-\sigma$
(c) Describe in your own words what $\mu$ and $\sigma$ tell you about the distribution.

Nick Johnson

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Problem 22

For a normal population of mean $0,$ show that the fraction of the population within one standard deviation of the mean does not depend on the standard deviation.
[Hint: Use the substitution $w=x / \sigma .]$

Ahmad Reda

Numerade Educator

Problem 23

Which of the following functions makes the most sense as a model for the probability density representing the time (in minutes, starting from $t=0$ ) that the next customer walks into a store?
(a) $p(t)=\left\{\begin{array}{ll}\cos t & 0 \leq t \leq 2 \pi \\ e^{t-2 \pi} & t \geq 2 \pi\end{array}\right.$
(b) $p(t)=3 e^{-3 t}$ for $t \geq 0$
(c) $p(t)=e^{-3 t}$ for $t \geq 0$
(d) $p(t)=1 / 4$ for $0 \leq t \leq 4$

Ahmad Reda

Numerade Educator

Problem 24

Let $P(x)$ be the cumulative distribution function for the household income distribution in the US in $2009 .^{11}$ Values of $P(x)$ are in the following table:
$$\begin{array}{l|c|c|c|c|c}
\hline \text { Income } x \text { (thousand } \$ & 20 & 40 & 60 & 75 & 100 \\
\hline P(x)(\%) & 29.5 & 50.1 & 66.8 & 76.2 & 87.1 \\
\hline
\end{array}$$
(a) What percent of the households made between $\$ 40,000$ and $\$ 60,000 ?$ More than $\$ 100,000 ?$
(b) Approximately what was the median income?
(c) Is the statement "More than one-third of households made between $\$ 40,000$ and $\$ 75,000 "$ true or false?

Ahmad Reda

Numerade Educator

Problem 25

If we think of an electron as a particle, the function $P(r)=1-\left(2 r^{2}+2 r+1\right) e^{-2 r}$ is the cumulative distribution function of the distance, $r,$ of the electron in a hydrogen atom from the center of the atom. The distance is measured in Bohr radii. (1 Bohr radius $=5.29 \times 10^{-11} \mathrm{m} .$ Niels Bohr $(1885-1962)$ was a Danish physicist.)
For example, $P(1)=1-5 e^{-2} \approx 0.32$ means that the electron is within 1 Bohr radius from the center of the atom $32 \%$ of the time.
(a) Find a formula for the density function of this distribution. Sketch the density function and the cumulative distribution function.
(b) Find the median distance and the mean distance. Near what value of $r$ is an electron most likely to be found?
(c) The Bohr radius is sometimes called the "radius of the hydrogen atom." Why?

Ahmad Reda

Numerade Educator

Problem 26

Explain what is wrong with the statement.
A median $T$ of a quantity distributed through a population satisfies $p(T)=0.5$ where $p$ is the density function.

Ahmad Reda

Numerade Educator

Problem 27

Explain what is wrong with the statement.
The following density function has median 1: $$p(x)=\left\{\begin{array}{lll}
0 & \text { for } & x<0 \\
2(1-x) & \text { for } & 0 \leq x \leq 1 \\
0 & \text { for } & x>1
\end{array}\right.$$

Ahmad Reda

Numerade Educator

Problem 28

Give an example of:
A distribution with a mean of $1 / 2$ and standard deviation $1 / 2$.

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Numerade Educator

Problem 29

Give an example of:
A distribution with a mean of $1 / 2$ and median $1 / 2$.

Ahmad Reda

Numerade Educator

Problem 30

A quantity $x$ is distributed through a population with probability density function $p(x)$ and cumulative distribution function $P(x) .$ Decide if each statement is true or false. Give an explanation for your answer.
If $p(10)=1 / 2,$ then half the population has $x<10$.

Ahmad Reda

Numerade Educator

Problem 31

A quantity $x$ is distributed through a population with probability density function $p(x)$ and cumulative distribution function $P(x) .$ Decide if each statement is true or false. Give an explanation for your answer.
If $P(10)=1 / 2,$ then half the population has $x<10$.

Ahmad Reda

Numerade Educator

Problem 32

A quantity $x$ is distributed through a population with probability density function $p(x)$ and cumulative distribution function $P(x) .$ Decide if each statement is true or false. Give an explanation for your answer.
If $p(10)=1 / 2,$ then the fraction of the population lying between $x=9.98$ and $x=10.04$ is about 0.03 .

Ahmad Reda

Numerade Educator

Problem 33

A quantity $x$ is distributed through a population with probability density function $p(x)$ and cumulative distribution function $P(x) .$ Decide if each statement is true or false. Give an explanation for your answer.
If $p(10)=p(20),$ then none of the population has $x$ values lying between 10 and 20 .

Ahmad Reda

Numerade Educator

Problem 34

A quantity $x$ is distributed through a population with probability density function $p(x)$ and cumulative distribution function $P(x) .$ Decide if each statement is true or false. Give an explanation for your answer.
If $P(10)=P(20),$ then none of the population has $x$ values lying between 10 and 20 .

Ahmad Reda

Numerade Educator