EXERCISES Section 5.5 (58-74)

58

A shipping company handles containers in three different sizes: (1) $27{\mathrm{ft}}^3\left(3\times 3\times 3\right)$ ,(2) $125{\mathrm{ft}}^3$ , and (3) $512{\mathrm{ft}}^3$ . Let $X_i\left(i=1,2,3\right)$ denote the number of type $i$ containers shipped during a given week. With ${\mu }_i=E\left(X_i\right)$ and ${\sigma }_i^2=V\left(X_i\right)$ , suppose that the mean values and standard deviations are as follows:

$${\mu }_1=200{\mu }_2=250{\mu }_3=100$$ $${\sigma }_1=10{\sigma }_2=12{\sigma }_3=8$$

a. Assuming that $X_1,X_2,X_3$ are independent, calculate the expected value and variance of the total volume shipped. [Hint: Volume $=27X_1+125X_2+512X_3$ .]

b. Would your calculations necessarily be correct if the $X_i$ ’s were not independent? Explain.

59

Let $X_1,X_2$ , and $X_3$ represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv’s with expected values ${\mu }_1,{\mu }_2$ , and ${\mu }_3$ and variances ${\sigma }_1^2,{\sigma }_2^2$ , and ${\sigma }_3^2$ , respectively.

a. If $\mu ={\mu }_2={\mu }_3=60$ and ${\sigma }_1^2={\sigma }_2^2={\sigma }_3^2=15$ , calculate $P\left(T_o\le 200\right)$ and $P\left(150\le T_o\le 200\right)$ ?

b. Using the ${\mu }_i$ ’s and ${\sigma }_i$ ’s given in part (a), calculate both $P\left(55\le \bar{X}\right)$ and $P\left(58\le \bar{X}\le 62\right)$ .

c. Using the ${\mu }_i$ ’s and ${\sigma }_i$ ’s given in part (a), calculate and interpret $P\left(-10\le X_1-.5X_2-.5X_3\le 5\right)$ .

d. If ${\mu }_1=40,{\mu }_2=50,{\mu }_3=60,{\sigma }_1^2=10,{\sigma }_2^2=12$ , and ${\sigma }_3^2=14$ , calculate $P\left(X_1+X_2+X_3\le 160\right)$ and also $P\left(X_1+X_2\ge 2X_3\right)$ .

60

Refer back to Example 5.31. Two cars with six-cylinder engines and three with four-cylinder engines are to be driven over a 300-mile course. Let $X_1,\dots X_5$ denote the resulting fuel efficiencies (mpg). Consider the linear combination

$$Y=\left(X_1+X_2\right)/2-\left(X_3+X_4+X_5\right)/3$$

which is a measure of the difference between four-cylinder and six-cylinder vehicles. Compute $P\left(0\le Y\right)$ and $P\left(Y>-2\right)$ . [Hint: $Y=a_1{X}_1+\cdots +a_5{X}_5$ , with $a_1=1/2,\dots ,a_5=-1/3\text{.}]$

61

Exercise 26 introduced random variables $X$ and $Y$ , the number of cars and buses, respectively, carried by a ferry on a single trip. The joint pmf of $X$ and $Y$ is given in the table in Exercise 7. It is readily verified that $X$ and $Y$ are independent.

a. Compute the expected value, variance, and standard deviation of the total number of vehicles on a single trip.

b. If each car is charged \ 3$andeachbus$$ {10}$ , compute the expected value, variance, and standard deviation of the revenue resulting from a single trip.

62

Manufacture of a certain component requires three different machining operations. Machining time for each operation has a normal distribution, and the three times are independent of one another. The mean values are $15,30$ , and $20\min$ , respectively, and the standard deviations are 1,2 , and 1.5 min, respectively. What is the probability that it takes at most 1 hour of machining time to produce a randomly selected component?

63

Refer to Exercise 3.

a. Calculate the covariance between $X_1=$ the number of customers in the express checkout and $X_2=$ the number of customers in the superexpress checkout.

b. Calculate $V\left(X_1+X_2\right)$ . How does this compare to $V\left(X_1\right)+V\left(X_2\right)$ ?

64

Suppose your waiting time for a bus in the morning is uniformly distributed on $\left[0,8\right]$ , whereas waiting time in the evening is uniformly distributed on $\left[0,10\right]$ independent of morning waiting time.

a. If you take the bus each morning and evening for a week, what is your total expected waiting time? [Hint: Define rv’s $X_1,\dots ,X_{10}$ and use a rule of expected value.]

b. What is the variance of your total waiting time?

c. What are the expected value and variance of the difference between morning and evening waiting times on a given day?

d. What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week?

65

Suppose that when the $\mathrm{pH}$ of a certain chemical compound is 5.00, the $\mathrm{pH}$ measured by a randomly selected beginning chemistry student is a random variable with mean 5.00 and standard deviation .2. A large batch of the compound is subdivided and a sample given to each student in a morning lab and each student in an afternoon lab. Let $\bar{X}=$ the average $\mathrm{pH}$ as determined by the morning students and $\bar{Y}=$ the average $\mathrm{pH}$ as determined by the afternoon students.

a. If $\mathrm{pH}$ is a normal variable and there are 25 students in each lab, compute $P\left(-.1\le \bar{X}-\bar{Y}\le .1\right)$ . [Hint: $\bar{X}-\bar{Y}$ is a linear combination of normal variables, so is normally distributed. Compute ${\mu }_{\bar{X}-\bar{Y}}$ and ${\sigma }_{\bar{X}-\bar{Y}}$ .]

b. If there are 36 students in each lab, but $\mathrm{pH}$ determinations are not assumed normal, calculate (approximately) $P\left(-.1\le \bar{X}-\bar{Y}\le .1\right)$ .

66

If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at 0 due to the loads is $a_1{X}_1+a_2{X}_2$ .

a. Suppose that $X_1$ and $X_2$ are independent rv’s with means 2 and 4 kips, respectively, and standard deviations .5 and $1.0\mathrm{kip}$ , respectively. If $a_1=5\mathrm{ft}$ and $a_2=10\mathrm{ft}$ , what is the expected bending moment and what is the standard deviation of the bending moment?

b. If $X_1$ and $X_2$ are normally distributed, what is the probability that the bending moment will exceed $75\mathrm{kip}-\mathrm{ft}$ ?

c. Suppose the positions of the two loads are random variables. Denoting them by $A_1$ and $A_2$ , assume that these variables have means of 5 and $10\mathrm{ft}$ , respectively, that each has a standard deviation of .5 , and that all $A_i$ ’s and $X_i$ ’s are independent of one another. What is the expected moment now?

d. For the situation of part (c), what is the variance of the bending moment?

e. If the situation is as described in part (a) except that $\mathrm{Corr}\left(X_1,X_2\right)=.5$ (so that the two loads are not independent), what is the variance of the bending moment?

67

One piece of PVC pipe is to be inserted inside another piece. The length of the first piece is normally distributed with mean value 20 in. and standard deviation .5 in. The length of the second piece is a normal rv with mean and standard deviation 15 in. and .4 in., respectively. The amount of overlap is normally distributed with mean value 1 in. and standard deviation .1 in. Assuming that the lengths and amount of overlap are independent of one another, what is the probability that the total length after insertion is between 34.5 in. and 35 in.?

68

Two airplanes are flying in the same direction in adjacent parallel corridors. At time $t=0$ , the first airplane is $10\mathrm{km}$ ahead of the second one. Suppose the speed of the first plane $\left(\mathrm{km}/\mathrm{hr}\right)$ is normally distributed with mean 520 and standard deviation 10 and the second plane’s speed is also normally distributed with mean and standard deviation 500 and 10 , respectively.

a. What is the probability that after $2\mathrm{hr}$ of flying, the second plane has not caught up to the first plane?

b. Determine the probability that the planes are separated by at most $10\mathrm{km}$ after $2\mathrm{hr}$ .

69

Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the number of cars coming from each road onto the freeway is a random variable, with expected value and standard deviation as given in the table.

	Road 1	Road 2	Road 3
Expected value	800	1000	600
Standard deviation	16	25	18

a. What is the expected total number of cars entering the freeway at this point during the period? [Hint: Let $X_i=$ the number from road $i$ .]

b. What is the variance of the total number of entering cars? Have you made any assumptions about the relationship between the numbers of cars on the different roads?

c. With $X_i$ denoting the number of cars entering from road $i$ during the period, suppose that $\mathrm{Cov}\left(X_1,X_2\right)=80$ , $\mathrm{Cov}\left(X_1,X_3\right)=90$ , and $\mathrm{Cov}\left(X_2,X_3\right)=100$ (so that the three streams of traffic are not independent). Compute the expected total number of entering cars and the standard deviation of the total.

70

Consider a random sample of size $n$ from a continuous distribution having median 0 so that the probability of any one observation being positive is .5 . Disregarding the signs of the observations, rank them from smallest to largest in absolute value, and let $W=$ the sum of the ranks of the observations having positive signs. For example, if the observations are $-.3,+.7,+2.1$ , and -2.5 , then the ranks of positive observations are 2 and 3 , so $W=5$ . In Chapter 15, $W$ will be called Wilcoxon’s signed-rank statistic. $W$ can be represented as follows:

$$\begin{array}{rl}W& =1\cdot Y_1+2\cdot Y_2+3\cdot Y_3+\cdots +n\cdot Y_n\\ & =\sum _{i=1}^{n}i\cdot Y_i\end{array}$$

where the $Y_i$ ’s are independent Bernoulli rv’s, each with $p=.5(Y_i=1$ corresponds to the observation with rank $i$ being positive).

a. Determine $E\left(Y_i\right)$ and then $E\left(W\right)$ using the equation for $W$ . [Hint: The first $n$ positive integers sum to $n\left(n+1\right)/2$ .]

b. Determine $V\left(Y_i\right)$ and then $V\left(W\right)$ . [Hint: The sum of the squares of the first $n$ positive integers can be expressed as $n\left(n+1\right)\left(2n+1\right)/6$ .]

71

In Exercise 66, the weight of the beam itself contributes to the bending moment. Assume that the beam is of uniform thickness and density so that the resulting load is uniformly distributed on the beam. If the weight of the beam is random, the resulting load from the weight is also random; denote this load by $W$ (kip-ft).

a. If the beam is $12\mathrm{ft}$ long, $W$ has mean 1.5 and standard deviation .25 , and the fixed loads are as described in part (a) of Exercise 66, what are the expected value and variance of the bending moment? [Hint: If the load due to the beam were $w$ kip-ft, the contribution to the bending moment would be $w{\int }_0^{12}xdx$ .]

b. If all three variables $(X_1,X_2$ , and $W)$ are normally distributed, what is the probability that the bending moment will be at most 200 kip-ft?

72

I have three errands to take care of in the Administration Building. Let $X_i=$ the time that it takes for the $i$ th errand $\left(i=1,2,3\right)$ , and let $X_4=$ the total time in minutes that I spend walking to and from the building and between each errand. Suppose the $X_i$ ’s are independent, and normally distributed, with the following means and standard deviations: ${\mu }_1=15,{\sigma }_1=4,{\mu }_2=5,{\sigma }_2=1,{\mu }_3=8$ , ${\sigma }_3=2,{\mu }_4=12,{\sigma }_4=3$ . I plan to leave my office at precisely 10:00 A.M. and wish to post a note on my door that reads,“I will return by $t$ A.M.” What time $t$ should I write down if I want the probability of my arriving after $t$ to be .01 ?

73

Suppose the expected tensile strength of type-A steel is $105\mathrm{ksi}$ and the standard deviation of tensile strength is 8 ksi. For type-B steel, suppose the expected tensile strength and standard deviation of tensile strength are $100\mathrm{ksi}$ and $6\mathrm{ksi}$ , respectively. Let $\bar{X}=$ the sample average tensile strength of a random sample of 40 type-A specimens, and let $\bar{Y}=$ the sample average tensile strength of a random sample of 35 type-B specimens.

a. What is the approximate distribution of $\bar{X}$ ? Of $\bar{Y}$ ?

b. What is the approximate distribution of $\bar{X}-\bar{Y}$ ? Justify your answer.

c. Calculate (approximately) $P\left(-1\le \bar{X}-\bar{Y}\le 1\right)$ .

d. Calculate $P\left(\bar{X}-\bar{Y}\ge 10\right)$ . If you actually observed

$\bar{X}-\bar{Y}\ge 10$ , would you doubt that ${\mu }_1-{\mu }_2=5$ ?

74

In an area having sandy soil, 50 small trees of a certain type were planted, and another 50 trees were planted in an area having clay soil. Let $X=$ the number of trees planted in sandy soil that survive 1 year and $Y=$ the number of trees planted in clay soil that survive 1 year. If the probability that a tree planted in sandy soil will survive 1 year is .7 and the probability of 1-year survival in clay soil is .6, compute an approximation to $P\left(-5\le X-Y\le 5\right)$ (do not bother with the continuity correction).