Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

Chapter 12

Probability and Calculus - all with Video Answers

Educators

Section 1

Discrete Random Variables

Problem 1

Table 4 is the probability table for a random variable $X .$ Find $\mathrm{E}(X), \operatorname{Var}(X),$ and the standard deviation of $X$.
$$\begin{aligned}&\begin{array}{lll} \text {Table 4} \\ \hline \text { Outcome } & 0 & 1 \\\text { Probability } & \frac{1}{5} & \frac{4}{5} \\\hline\end{array}\end{aligned}$$

Kaitlin Yaeger

Kaitlin Yaeger

Numerade Educator

Problem 2

Find $\mathrm{E}(X), \operatorname{Var}(X),$ and the standard deviation of $X,$ where $X$ is the random variable whose probability table is given in Table 5.
$$\begin{array}{llll} \text {Table 5} \\ \hline \text { Outcome } & 1 & 2 & 3 \\\text { Probability } & \frac{4}{9} & \frac{4}{9} & \frac{1}{9} \\\hline\end{array}$$

Kaitlin Yaeger

Numerade Educator

Problem 3

Compute the variances of the three random variables whose probability tables are given in Table $6 .$ Relate the sizes of the variances to the spread of the values of the random variable.
$$\begin{array}{llc} \text {Table 6} \\ \hline & \text { Outcome } & \text { Probability } \\\hline & 4 & .5 \\(\mathrm{a}) & 6 & .5 \\\hline & 3 & .5 \\(\mathrm{b}) & 7 & .5 \\\hline & 1 & .5 \\(\mathrm{c}) & 9 & .5 \\\hline \end{array}$$

Kaitlin Yaeger

Numerade Educator

Problem 4

Compute the variances of the two random variables whose probability tables are given in Table $7 .$ Relate the sizes of the variances to the spread of the values of the random variables.
$$\begin{array}{cc} \text {Table 7} \\ \hline & \text { Outcome } & \text { Probability } \\\hline & 2 & .1 \\\text { (a) } & 4 & .4 \\& 6 & .4 \\& 8 & .1 \\\hline & 2 & .3 \\\text { (b) } & 4 & .2 \\& 6 & .2 \\& 8 & .3 \\ \hline\end{array}$$

Kaitlin Yaeger

Numerade Educator

Problem 5

The number of accidents per week at a busy intersection was recorded for a year. There were 11 weeks with no accidents, 26 weeks with one accident, 13 weeks with two accidents, and 2 weeks with three accidents. A week is to be selected at random and the number of accidents noted. Let $X$ be the outcome. Then, $X$ is a random variable taking on the values $0,1,2,$ and $3.$
(a) Write out a probability table for $X$.
(b) Compute $E(X).$
(c) Interpret $\mathrm{E}(X).$

Kaitlin Yaeger

Numerade Educator

Problem 6

The number of phone calls coming into a telephone switchboard during each minute was recorded during an entire hour. During 30 of the 1-minute intervals there were no calls, during 20 intervals there was one call, and during 10 intervals there were two calls. A 1-minute interval is to be selected at random and the number of calls noted. Let $X$ be the outcome. Then, $X$ is a random variable taking on the values $0,1,$ and $2.$
(a) Write out a probability table for $X.$
(b) Compute $E(X).$
(c) Interpret $E(X).$

Kaitlin Yaeger

Numerade Educator

Problem 7

Consider a circle with radius 1.
(a) What percentage of the points lies within $\frac{1}{2}$ unit of the center?
(b) Let $c$ be a constant with $0<c<1 .$ What percentage of the points lies within $c$ unit of the center?

Stanley Enemuo

Numerade Educator

Problem 8

Consider a circle with circumference 1. An arrow (or spinner) is attached at the center so that, when flicked, it spins freely. Upon stopping, it points to a particular point on the circumference of the circle. Determine the likelihood that the point is
(a) On the top half of the circumference.
(b) On the top quarter of the circumference.
(c) On the top one-hundredth of the circumference.
(d) Exactly at the top of the circumference.

Stanley Enemuo

Numerade Educator

Problem 9

A citrus grower anticipates a profit of 100,000 dollars this year if the nightly temperatures remain mild. Unfortunately, the weather forecast indicates a $25 \%$ chance that the temperatures will drop below freezing during the next week. Such freezing weather will destroy $40 \%$ of the crop and reduce the profit to 60,000 dollars. However, the grower can protect the citrus fruit against the possible freezing (using smudge pots, electric fans, and so on) at a cost of $\$5000.$ Should the grower spend the $\$ 5000$ and thereby reduce the profit to $\$ 95,000 ?$ [Hint: Compute $E(X)$, where $X$ is the profit the grower will get if he does nothing to protect the fruit.]

Stanley Enemuo

Numerade Educator

Problem 10

Suppose that the weather forecast in Exercise 9 indicates a $10 \%$ chance that cold weather will reduce the citrus grower's profit from $\$ 100,000$ to $\$ 85,000$ and a $10 \%$ chance that cold weather will reduce the profit to $\$ 75,000$. Should the grower spend $\$ 5000$ to protect the citrus fruit against the possible bad weather?

Stanley Enemuo

Numerade Educator