Example 5.25
PROPOSITION
Let X
1
, X
2
, . . . , X
n
be a random sample from a normal distribution with mean m
and standard deviation s. Then for any n, is normally distributed with mean m and standard deviation
, as is T
o
with mean nm and standard deviation .
1ns s
1n X
We know everything there is to know about the and T
o
distributions when the pop- ulation distribution is normal. In particular, probabilities such as Pa b and
P c T
o
d can be obtained simply by standardizing. Figure 5.14 illustrates the
proposition. X
X
X distribution when n ⴝ 10
X distribution when n ⴝ 4
Population distribution
Figure 5.14 A normal population distribution and sampling distributions
X
The time that it takes a randomly selected rat of a certain subspecies to find its way through a maze is a normally distributed rv with m 1.5 min and s .35 min.
Suppose five rats are selected. Let X
1
, . . . , X
5
denote their times in the maze. Assuming the X
i
’s to be a random sample from this normal distribution, what is the probability that the total time T
o
X
1
. . . X
5
for the five is between 6 and 8 min? By the proposition, T
o
has a normal distribution with m
To
nm 51.5 7.5
and variance , so s
To
.783. To standardize T
o
, subtract m
To
and divide by s
To
:
P 1.92 Z .64 .64 1.92 .7115
Determination of the probability that the sample average time a normally
distributed variable is at most 2.0 min requires m m 1.5 and . Then
■ P
X 2.0 5 P aZ
2.0 2 1.5 .1565
b 5 PZ 3.19 5 3.19 5 .9993 .35 15 5 .1565
s
X
5 s 1n 5
X
X P
6 T
o
8 5 P a
6 2 7.5 .783
Z 8 2 7.5
.783 b s
T
o
2
5 ns
2
5 5.1225 5 .6125
A proof of the result for T
o
when n 2 is possible using the method in Example 5.21, but the details are messy. The general result is usually proved using a theoretical tool called a moment generating
function . One of the chapter references can be consulted for more information.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Example 5.26
THEOREM The Central Limit Theorem CLT
Let X
1
, X
2
, . . . , X
n
be a random sample from a distribution with mean m and variance s
2
. Then if n is sufficiently large, has approximately a normal dis- tribution with m m and ,
and T
o
also has approximately a normal s
X 2
5 s
2
n
X
X
Figure 5.15 illustrates the Central Limit Theorem. According to the CLT, when n is large and we wish to calculate a probability such as Pa b, we need only
“pretend” that is normal, standardize it, and use the normal table. The resulting answer will be approximately correct. The exact answer could be obtained only by
first finding the distribution of , so the CLT provides a truly impressive shortcut. The proof of the theorem involves much advanced mathematics.
X X
X
X distribution for
small to moderate n Population
distribution X
distribution for large n approximately normal
Figure 5.15
The Central Limit Theorem illustrated
The Central Limit Theorem
When the X
i
’s are normally distributed, so is for every sample size n. The deri- vations in Example 5.20 and simulation experiment of Example 5.23 suggest that
even when the population distribution is highly nonnormal, averaging produces a distribution more bell-shaped than the one being sampled. A reasonable conjecture
is that if n is large, a suitable normal curve will approximate the actual distribu- tion of . The formal statement of this result is the most important theorem of
probability.
X X
The amount of a particular impurity in a batch of a certain chemical product is a random variable with mean value 4.0 g and standard deviation 1.5 g. If 50 batches
are independently prepared, what is the approximate probability that the sample average amount of impurity is between 3.5 and 3.8 g? According to the rule of
thumb to be stated shortly, n 50 is large enough for the CLT to be applicable. then has approximately a normal distribution with mean value m 4.0 and
, so
.94 2.36 .1645 ■
P 3.5 X 3.8 P
a 3.5 2 4.0
.2121 Z
3.8 2 4.0 .2121 b
s
X
5 1.5 150 5 .2121
X
X X
distribution with m
To
nm ,
. The larger the value of n, the better the approximation.
s
T 2
o
5 ns
2
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Example 5.28
A certain consumer organization customarily reports the number of major defects for each new automobile that it tests. Suppose the number of such defects for a certain
model is a random variable with mean value 3.2 and standard deviation 2.4. Among 100 randomly selected cars of this model, how likely is it that the sample average
number of major defects exceeds 4? Let X
i
denote the number of major defects for the ith car in the random sample. Notice that X
i
is a discrete rv, but the CLT is appli- cable whether the variable of interest is discrete or continuous. Also, although the
fact that the standard deviation of this nonnegative variable is quite large relative to the mean value suggests that its distribution is positively skewed, the large sample
size implies that does have approximately a normal distribution. Using m 3.2 and s .24,
■ The CLT provides insight into why many random variables have probability distri-
butions that are approximately normal. For example, the measurement error in a sci- entific experiment can be thought of as the sum of a number of underlying
perturbations and errors of small magnitude.
A practical difficulty in applying the CLT is in knowing when n is sufficiently large. The problem is that the accuracy of the approximation for a particular n
depends on the shape of the original underlying distribution being sampled. If the underlying distribution is close to a normal density curve, then the approximation
will be good even for a small n, whereas if it is far from being normal, then a large n
will be required. P
X . 4 P aZ .
4 2 3.2 .24 b
5 1 2 3.33 5 .0004
X X
X
Rule of Thumb If n 30, the Central Limit Theorem can be used.
There are population distributions for which even an n of 40 or 50 does not suffice, but such distributions are rarely encountered in practice. On the other hand, the rule
of thumb is often conservative; for many population distributions, an n much less than 30 would suffice. For example, in the case of a uniform population distribution,
the CLT gives a good approximation for n 12.
Consider the distribution shown in Figure 5.16 for the amount purchased rounded to the nearest dollar by a randomly selected customer at a particular gas station a
similar distribution for purchases in Britain in £ appeared in the article “Data Mining for Fun and Profit,” Statistical Science, 2000: 111–131; there were big
spikes at the values, 10, 15, 20, 25, and 30. The distribution is obviously quite non-normal.
We asked Minitab to select 1000 different samples, each consisting of n 15 observations, and calculate the value of the sample mean
for each one. Figure 5.17 is a histogram of the resulting 1000 values; this is the approximate sam-
pling distribution of under the specified circumstances. This distribution is clearly approximately normal even though the sample size is actually much smaller than
X X
Example 5.27
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Figure 5.16 Probability distribution of
X amount of gasoline purchased
0.16
Probability
Purchase amount
0.14 0.12
0.10 0.08
0.06 0.04
0.02 0.00
60 55
50 45
40 35
30 25
20 15
10 5
Density
Mean
0.14 0.12
0.10 0.08
0.06 0.04
0.02 0.00
36 33
30 27
24 21
18
Figure 5.17
Approximate sampling distribution of the sample mean amount purchased when n 15 and the population distribution is as shown in Figure 5.16
30, our rule-of-thumb cutoff for invoking the Central Limit Theorem. As further evidence for normality, Figure 5.18 shows a normal probability plot of the 1000
values; the linear pattern is very prominent. It is typically not non-normality in the central part of the population distribution that causes the CLT to fail, but instead
very substantial skewness. x
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Other Applications of the Central Limit Theorem
The CLT can be used to justify the normal approximation to the binomial distribu- tion discussed in Chapter 4. Recall that a binomial variable X is the number of suc-
cesses in a binomial experiment consisting of n independent successfailure trials with p PS for any particular trial. Define a new rv X
1
by
and define X
2
, X
3
, . . . , X
n
analogously for the other n 1 trials. Each X
i
indicates whether or not there is a success on the corresponding trial.
Because the trials are independent and PS is constant from trial to trial, the X
i
’s are iid a random sample from a Bernoulli distribution. The CLT then implies that if n is sufficiently large, both the sum and the average of the X
i
’s have approxi- mately normal distributions. When the X
i
’s are summed, a 1 is added for every S that occurs and a 0 for every F, so X
1
. . . X
n
X . The sample mean of the X
i
’s is X
n, the sample proportion of successes. That is, both X and Xn are approximately normal when n is large. The necessary sample size for this approximation depends
on the value of p: When p is close to .5, the distribution of each X
i
is reasonably sym- metric see Figure 5.19, whereas the distribution is quite skewed when p is near 0
or 1. Using the approximation only if both np 10 and n1 p 10 ensures that n
is large enough to overcome any skewness in the underlying Bernoulli distribution. X
1
5 e
1 if the 1st trial results in a success
if the 1st trial results in a failure
Mean
Mean StDev
N RJ
P-Value 26.49
3.112 1000
0.999 0.100
99.99
95 80
50 20
5 1
0.01
P er
cent
99
40 35
30 25
20 15
Figure 5.18 Normal probability plot from Minitab of the 1000 values based on samples
of size n 15
x
Recall from Section 4.5 that X has a lognormal distribution if lnX has a nor- mal distribution.
■
1 a
1 b
Figure 5.19
Two Bernoulli distributions: a p .4 reasonably symmetric; b p .1
very skewed
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
PROPOSITION
Let X
1
, X
2
, . . . , X
n
be a random sample from a distribution for which only pos- itive values are possible [PX
i
0 1]. Then if n is sufficiently large, the product Y X
1
X
2
. . . X
n
has approximately a lognormal distribution.
To verify this, note that lnY lnX
1
lnX
2
. . . lnX
n
Since lnY is a sum of independent and identically distributed rv’s [the lnX
i
s], it is approximately normal when n is large, so Y itself has approximately a lognormal dis-
tribution. As an example of the applicability of this result, Bury Statistical Models in Applied Science,
Wiley, p. 590 argues that the damage process in plastic flow and crack propagation is a multiplicative process, so that variables such as percentage
elongation and rupture strength have approximately lognormal distributions.
EXERCISES
Section 5.4 46–57
46.
The inside diameter of a randomly selected piston ring is a random variable with mean value 12 cm and standard
deviation .04 cm. a.
If is the sample mean diameter for a random sample of n
16 rings, where is the sampling distribution of centered, and what is the standard deviation of the
distribution?
b.
Answer the questions posed in part a for a sample size of n 64 rings.
c.
For which of the two random samples, the one of part a or the one of part b, is more likely to be within .01 cm
of 12 cm? Explain your reasoning.
47.
Refer to Exercise 46. Suppose the distribution of diameter is normal.
a.
Calculate P11.99 12.01 when n 16.
b.
How likely is it that the sample mean diameter exceeds 12.01 when n 25?
48.
The National Health Statistics Reports dated Oct. 22, 2008, stated that for a sample size of 277 18-year-old American
males, the sample mean waist circumference was 86.3 cm. A somewhat complicated method was used to estimate various
population percentiles, resulting in the following values:
5
th
10
th
25
th
50
th
75
th
90
th
95
th
69.6 70.9
75.2 81.3
95.4 107.1
116.4
a.
Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning. If your
answer is no, conjecture the shape of the population dis- tribution.
b.
Suppose that the population mean waist size is 85 cm and that the population standard deviation is 15 cm.
How likely is it that a random sample of 277 individu- als will result in a sample mean waist size of at least
86.3 cm? X
X X
X X
c.
Referring back to b, suppose now that the population mean waist size in 82 cm. Now what is the approxi-
mate probability that the sample mean will be at least 86.3 cm? In light of this calculation, do you think that
82 cm is a reasonable value for m?
49.
There are 40 students in an elementary statistics class. On the basis of years of experience, the instructor knows that
the time needed to grade a randomly chosen first examina- tion paper is a random variable with an expected value of 6
min and a standard deviation of 6 min. a.
If grading times are independent and the instructor begins grading at 6:50
P
.
M
. and grades continuously, what is the approximate probability that he is through
grading before the 11:00
P
.
M
. TV news begins?
b.
If the sports report begins at 11:10, what is the probabil- ity that he misses part of the report if he waits until grad-
ing is done before turning on the TV?
50.
The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi.
a.
What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between
9900 and 10,200?
b.
If the sample size had been 15 rather than 40, could the probability requested in part a be calculated from the
given information?
51.
The time taken by a randomly selected applicant for a mort- gage to fill out a certain form has a normal distribution with
mean value 10 min and standard deviation 2 min. If five individuals fill out a form on one day and six on another,
what is the probability that the sample average amount of time taken on each day is at most 11 min?
52.
The lifetime of a certain type of battery is normally distrib- uted with mean value 10 hours and standard deviation
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
DEFINITION
1 hour. There are four batteries in a package. What lifetime value is such that the total lifetime of all batteries in a pack-
age exceeds that value for only 5 of all packages?
53.
Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.2.
a.
If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 9 pins
is at least 51?
b.
Without assuming population normality, what is the approximate probability that the sample mean hard-
ness for a random sample of 40 pins is at least 51?
54.
Suppose the sediment density gcm of a randomly selected specimen from a certain region is normally distributed with
mean 2.65 and standard deviation .85 suggested in “Modeling Sediment and Water Column Interactions for
Hydrophobic Pollutants,” Water Research, 1984: 1169–1174. a.
If a random sample of 25 specimens is selected, what is the probability that the sample average sediment density
is at most 3.00? Between 2.65 and 3.00?
b.
How large a sample size would be required to ensure that the first probability in part a is at least .99?
55.
The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter
m 50. What is the approximate probability that
5.5
The Distribution of a Linear Combination
The sample mean and sample total T
o
are special cases of a type of random vari- able that arises very frequently in statistical applications.
X
a.
Between 35 and 70 tickets are given out on a particular day? [Hint: When m is large, a Poisson rv has approxi-
mately a normal distribution.]
b.
The total number of tickets given out during a 5-day week is between 225 and 275?
56.
A binary communication channel transmits a sequence of “bits” 0s and 1s. Suppose that for any particular bit trans-
mitted, there is a 10 chance of a transmission error a 0 becoming a 1 or a 1 becoming a 0. Assume that bit errors
occur independently of one another. a.
Consider transmitting 1000 bits. What is the approximate probability that at most 125 transmission errors occur?
b.
Suppose the same 1000-bit message is sent two different times independently of one another. What is the approx-
imate probability that the number of errors in the first transmission is within 50 of the number of errors in the
second?
57.
Suppose the distribution of the time X in hours spent by students at a certain university on a particular project is
gamma with parameters a 50 and b 2. Because a is large, it can be shown that X has approximately a normal
distribution. Use this fact to compute the approximate prob- ability that a randomly selected student spends at most 125
hours on the project.
Given a collection of n random variables X
1
, . . . , X
n
and n numerical constants a
1
, . . . , a
n
, the rv 5.7
is called a linear combination of the X
i
’s. Y 5 a
1
X
1
1 c 1 a
n
X
n
5
g
n i5
1
a
i
X
i
For example, 4X
1
5X
2
8X
3
is a linear combination of X
1
, X
2
, and X
3
with a
1
4, a
2
5, and a
3
8. Taking a
1
a
2
. . . a
n
1 gives Y X
1
. . . X
n
T
o
, and yields
Notice that we are not requiring the X
i
’s to be independent or identically distrib- uted. All the X
i
’s could have different distributions and therefore different mean values and variances. We first consider the expected value and variance of a lin-
ear combination. Y 5
1 n
X
1
1 c 1 1
n X
n
5 1
n X
1
1 c 1 X
n
5 1
n T
o
5 X a
1
5 a
2
5 c 5 a
n
5
1 n
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
PROPOSITION Let X
1
, X
2
, . . . , X
n
have mean values m
1
, . . . , m
n
, respectively, and variances , respectively.
1.
Whether or not the X
i
’s are independent, E
a
1
X
1
a
2
X
2
. . . a
n
X
n
a
1
E X
1
a
2
E X
2
. . . a
n
E X
n
a
1
m
1
. . . a
n
m
n
5.8
2.
If X
1
, . . . , X
n
are independent, 5.9
and 5.10
3.
For any X
1
, . . . , X
n
, 5.11
V a
1
X
1
1 c 1 a
n
X
n
5
g
n i5
1
g
n j5
1
a
i
a
j
CovX
i
, X
j
s
a
1
X
1
1c1a
n
X
n
5 1a
1 2
s
1 2
1 c 1 a
n 2
s
n 2
5 a
1 2
s
1 2
1 c 1 a
n 2
s
n 2
V a
1
X
1
1 a
2
X
2
1 c 1 a
n
X
n
5 a
1 2
V X
1
1 a
2 2
V X
2
1 c1 a
n 2
V X
n
s
1 2
, c
, s
n 2
Proofs are sketched out at the end of the section. A paraphrase of 5.8 is that the expected value of a linear combination is the same as the linear combination of the
expected values—for example, E2X
1
5X
2
2m
1
5m
2
. The result 5.9 in Statement 2 is a special case of 5.11 in Statement 3; when the X
i
’s are independ- ent, CovX
i
, X
j
0 for i ⬆ j and VX
i
for i j this simplification actually occurs when the X
i
’s are uncorrelated, a weaker condition than independence. Specializing to the case of a random sample X
i
’s iid with a
i
1n for every i gives E m and V s
2
n, as discussed in Section 5.4. A similar comment applies to the rules for T
o
. A gas station sells three grades of gasoline: regular, extra, and super. These are
priced at 3.00, 3.20, and 3.40 per gallon, respectively. Let X
1
, X
2
, and X
3
denote the amounts of these grades purchased gallons on a particular day. Suppose the X
i
’s are independent with m
1
1000, m
2
500, m
3
300, s
1
100, s
2
80, and s
3
50. The revenue from sales is Y 3.0X
1
3.2X
2
3.4X
3
, and E
Y 3.0m
1
3.2m
2
3.4m
3
5620 ■
The Difference Between Two Random Variables
An important special case of a linear combination results from taking n 2, a
1
1, and a
2
1: Y a
1
X
1
a
2
X
2
X
1
X
2
We then have the following corollary to the proposition. s
Y
5 1184,436 5 429.46
V Y 5 3.0
2
s
1 2
1 3.2
2
s
2 2
1 3.4
2
s
3 2
5 184,436
X X
Example 5.29
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Example 5.30
COROLLARY
E X
1
X
2
EX
1
EX
2
for any two rv’s X
1
and X
2
. VX
1
X
2
V X
1
VX
2
if X
1
and X
2
are independent rv’s.
The expected value of a difference is the difference of the two expected values, but the variance of a difference between two independent variables is the sum, not the
difference, of the two variances. There is just as much variability in X
1
X
2
as in X
1
X
2
[writing X
1
X
2
X
1
1X
2
, 1X
2
has the same amount of variability as X
2
itself]. A certain automobile manufacturer equips a particular model with either a six-cylinder
engine or a four-cylinder engine. Let X
1
and X
2
be fuel efficiencies for independently and randomly selected six-cylinder and four-cylinder cars, respectively. With m
1
22, m
2
26, s
1
1.2, and s
2
1.5, E
X
1
X
2
m
1
m
2
22 26 4
If we relabel so that X
1
refers to the four-cylinder car, then EX
1
X
2
4, but the variance of the difference is still 3.69.
■
The Case of Normal Random Variables
When the X
i
’s form a random sample from a normal distribution, and T
o
are both normally distributed. Here is a more general result concerning linear combinations.
X s
X
1
2X
2
5 13.69 5 1.92
V X
1
2 X
2
5 s
1 2
1 s
2 2
5 1.2
2
1 1.5
2
5 3.69
Example 5.31
Example 5.29 continued
PROPOSITION If X
1
, X
2
, . . . , X
n
are independent, normally distributed rv’s with possibly dif- ferent means andor variances, then any linear combination of the X
i
’s also has a normal distribution. In particular, the difference X
1
X
2
between two independent, normally distributed variables is itself normally distributed.
The total revenue from the sale of the three grades of gasoline on a particular day was Y 3.0X
1
3.2X
2
3.4X
3
, and we calculated m
Y
5620 and assuming inde- pendence s
Y
429.46. If the X
i
s are normally distributed, the probability that rev- enue exceeds 4500 is
■ The CLT can also be generalized so it applies to certain linear combinations.
Roughly speaking, if n is large and no individual term is likely to contribute too much to the overall value, then Y has approximately a normal distribution.
Proofs for the Case n 2
For the result concerning expected values, suppose that X
1
and X
2
are continuous with joint pdf fx
1
, x
2
. Then 5 PZ . 22.61 5 1 2 22.61 5 .9955
PY . 4500 5 P aZ .
4500 2 5620 429.46
b
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
a
1
E X
1
a
2
E X
2
Summation replaces integration in the discrete case. The argument for the variance result does not require specifying whether either variable is discrete or continuous.
Recalling that VY E[Y m
Y 2
], V
a
1
X
1
a
2
X
2
E{[a
1
X
1
a
2
X
2
a
1
m
1
a
2
m
2
]
2
} The expression inside the braces is a linear combination of the variables Y
1
X
1
m
1 2
, Y
2
X
2
m
2 2
, and Y
3
X
1
m
1
X
2
m
2
, so carrying the E oper- ation through to the three terms gives
as required.
■ a
1 2
V X
1
1 a
2 2
V X
2
1 2a
1
a
2
CovX
1
, X
2
5 E 5a
1 2
X
1
2 m
1 2
1 a
2 2
X
2
2 m
2 2
1 2a
1
a
2
X
1
2 m
1
X
2
2 m
2
6 5 a
1
冮
` 2`
x
1
f
X
1
x
1
dx
1
1 a
2
冮
` 2`
x
2
f
X
2
x
2
dx
2
1 a
2
冮
` 2`
冮
` 2`
x
2
f x
1
, x
2
dx
1
dx
2
5 a
1
冮
` 2`
冮
` 2`
x
1
f x
1
, x
2
dx
2
dx
1
Ea
1
X
1
1 a
2
X
2
5
冮
` 2`
冮
` 2`
a
1
x
1
1 a
2
x
2
fx
1
, x
2
dx
1
dx
2
EXERCISES
Section 5.5 58–74
58.
A shipping company handles containers in three different sizes: 1 27 ft
3
3
3
3, 2 125 ft
3
, and 3 512 ft
3
. Let X
i
i 1, 2, 3 denote the number of type i containers shipped during a given week. With m
i
E X
i
and , suppose that the mean values and standard
deviations are as follows:
m
1
200 m
2
250 m
3
100 s
1
10 s
2
12 s
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 m
1
, m
2
, and m
3
and variances , and , respec-
tively. a.
If m m
2
m
3
60 and , calculate
P
T
o
200 and P150
T
o
200?
b.
Using the m
i
’
s and s
i
’
s given in part a, calculate both P
55 and P58 62.
c.
Using the m
i
’
s and s
i
’
s given in part a, calculate and interpret P10 X
1
.5X
2
.5X
3
5.
d.
If m
1
40, m
2
50, m
3
60, , and
s
1 2
5 10, s
2 2
5 12
X X
s
1 2
5 s
2 2
5 s
3 2
5 15
s
3 2
s
1 2
, s
2 2
s
i 2
5 V X
i
, calculate PX
1
X
2
X
3
160 and also P
X
1
X
2
2X
3
.
60.
Five automobiles of the same type are to be driven on a 300- mile trip. The first two will use an economy brand of gaso-
line, and the other three will use a name brand. Let X
1
, X
2
, X
3
, X
4
, and X
5
be the observed fuel efficiencies mpg for the five cars. Suppose these variables are independent and nor-
mally distributed with m
1
m
2
20, m
3
m
4
m
5
21, and s
2
4 for the economy brand and 3.5 for the name brand. Define an rv Y by
so that Y is a measure of the difference in efficiency between economy gas and name-brand gas. Compute P0 Y and
P 1 Y 1. [Hint: Y a
1
X
1
. . . a
5
X
5
, with .]
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 de- viation of the total number of vehicles on a single trip.
a
1
5
1 2
, c
, a
5
5 2
1 3
Y 5 X
1
1 X
2
2 2
X
3
1 X
4
1 X
5
3
s
3 2
5 14
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
b.
If each car is charged 3 and each bus 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 indepen- dent 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 VX
1
X
2
. How does this compare to VX
1
V X
2
?
64.
Suppose your waiting time for a bus in the morning is uni- formly distributed on [0, 8], whereas waiting time in the
evening is uniformly distributed on [0, 10] 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
, . . . , 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 differ- ence between morning and evening waiting times on a
given day?
d.
What are the expected value and variance of the differ- ence between total morning waiting time and total
evening waiting time for a particular week?
65.
Suppose that when the pH of a certain chemical compound is 5.00, the pH measured by a randomly selected begin-
ning chemistry student is a random variable with mean 5.00 and standard deviation .2. A large batch of the com-
pound is subdivided and a sample given to each student in a morning lab and each student in an afternoon lab. Let
the average pH as determined by the morning stu- dents and the average pH as determined by the after-
noon students. a.
If pH is a normal variable and there are 25 students in each lab, compute P.1 .1. [Hint: is
a linear combination of normal variables, so is normally distributed. Compute
and .]
s
X2Y
m
X2Y
Y X
Y X
Y X
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 kip, respectively. If a
1
5 ft and a
2
10 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 proba- bility that the bending moment will exceed 75 kip-ft?
c.
Suppose the positions of the two loads are random vari- ables. Denoting them by A
1
and A
2
, assume that these variables have means of 5 and 10 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 CorrX
1
, X
2
.5 so that the two loads are not inde- pendent, 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 km
ahead of the second one. Suppose the speed of the first plane kmhr 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 hr of flying, the sec- ond plane has not caught up to the first plane?
b.
Determine the probability that the planes are separated by at most 10 km after 2 hr.
69.
Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the num-
ber of cars coming from each road onto the freeway is a ran- dom 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 rela-
tionship between the numbers of cars on the different roads?
0, 1 x, 1 ⴚ x
y
b.
If there are 36 students in each lab, but pH determina- tions are not assumed normal, calculate approximately
P .1 .1.
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
. Y
X
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
c.
With X
i
denoting the number of cars entering from road i
during the period, suppose that CovX
1
, X
2
80, CovX
1
, X
3
90, and CovX
2
, X
3
100 so that the three streams of traffic are not independent. Compute
the expected total number of entering cars and the stan- dard deviation of the total.
70.
Consider a random sample of size n from a continuous dis- tribution 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: W
1 Y
1
2 Y
2
3 Y
3
. . . n Y
n
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 EY
i
and then EW using the equation for W. [Hint: The first n positive integers sum to nn 12.]
b.
Determine VY
i
and then VW. [Hint: The sum of the squares of the first n positive integers can be expressed
as nn 12n 16.]
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 ran-
dom, the resulting load from the weight is also random; denote this load by W kip-ft.
a.
If the beam is 12 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 vari- ance of the bending moment? [Hint: If the load due to the
5
g
n i5
1
i Y
i
beam were w kip-ft, the contribution to the bending moment would be w
冕
.]
b.
If all three variables X
1
, X
2
, and W are normally distrib- uted, 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 ith errand i 1, 2, 3, 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 dis- tributed, with the following means and standard deviations:
m
1
15, s
1
4, m
2
5, s
2
1, m
3
8, s
3
2, m
4
12, s
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 proba- bility of my arriving after t to be .01?
73.
Suppose the expected tensile strength of type-A steel is 105 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 ksi and
6 ksi, respectively. Let the sample average tensile strength of a random sample of 40 type-A specimens, and
let the sample average tensile strength of a random sample of 35 type-B specimens.
a.
What is the approximate distribution of ? Of ?
b.
What is the approximate distribution of ? Justify your answer.
c.
Calculate approximately P1 1.
d.
Calculate P 10. If you actually observed 10, would you doubt that m
1
m
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 P5 X Y 5 do not bother with
the continuity correction. Y
X Y
X Y
X Y
X Y
X Y
X
12
x dx
SUPPLEMENTARY EXERCISES
75–96
75.
A restaurant serves three fixed-price dinners costing 12, 15, and 20. For a randomly selected couple dining at this
restaurant, let X the cost of the man’s dinner and Y the cost of the woman’s dinner. The joint pmf of X and Y is
given in the following table:
a.
Compute the marginal pmf’s of X and Y.
b.
What is the probability that the man’s and the woman’s dinner cost at most 15 each?
c.
Are X and Y independent? Justify your answer.
d.
What is the expected total cost of the dinner for the two people?
e.
Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find the message “You will
receive as a refund the difference between the cost of the more expensive and the less expensive meal that you have
chosen.” How much would the restaurant expect to refund? y
p x, y
12 15
20 12
.05 .05
.10 x
15 .05
.10 .35
20 .20
.10
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
76.
In cost estimation, the total cost of a project is the sum of component task costs. Each of these costs is a random
variable with a probability distribution. It is customary to obtain information about the total cost distribution by
adding together characteristics of the individual compo- nent cost distributions—this is called the “roll-up” proce-
dure. For example, EX
1
. . . X
n
EX
1
. . . E
X
n
, so the roll-up procedure is valid for mean cost. Suppose that there are two component tasks and that X
1
and X
2
are independent, normally distributed random vari- ables. Is the roll-up procedure valid for the 75th per-
centile? That is, is the 75th percentile of the distribution of X
1
X
2
the same as the sum of the 75th percentiles of the two individual distributions? If not, what is the rela-
tionship between the percentile of the sum and the sum of percentiles? For what percentiles is the roll-up procedure
valid in this case?
77.
A health-food store stocks two different brands of a certain type of grain. Let X the amount lb of brand A on hand
and Y the amount of brand B on hand. Suppose the joint pdf of X and Y is
a.
Draw the region of positive density and determine the value of k.
b.
Are X and Y independent? Answer by first deriving the marginal pdf of each variable.
c.
Compute PX Y 25.
d.
What is the expected total amount of this grain on hand?
e.
Compute CovX, Y and CorrX, Y.
f.
What is the variance of the total amount of grain on hand?
78.
The article “Stochastic Modeling for Pavement Warranty Cost Estimation” J. of Constr. Engr. and Mgmnt., 2009:
352–359 proposes the following model for the distribution of Y time to pavement failure. Let X
1
be the time to fail- ure due to rutting, and X
2
be the time to failure due to trans- verse cracking; these two rvs are assumed independent.
Then Y minX
1
, X
2
. The probability of failure due to either one of these distress modes is assumed to be an
increasing function of time t. After making certain distribu- tional assumptions, the following form of the cdf for each
mode is obtained:
where is the standard normal cdf. Values of the five parameters a, b, c, d, and e are 25.49, 1.15, 4.45,
1.78, and .171 for cracking and 21.27, .0325, .972, .00028, and .00022 for rutting. Determine the probabil-
ity of pavement failure within t 5 years and also t
10 years.
79.
Suppose that for a certain individual, calorie intake at breakfast is a random variable with expected value 500
and standard deviation 50, calorie intake at lunch is c a 1 bt c 1 dt 1 et
2 12
d
f x, y 5
e kxy
x 0, y 0, 20 x 1 y 30
otherwise
random with expected value 900 and standard deviation 100, and calorie intake at dinner is a random variable
with expected value 2000 and standard deviation 180. Assuming that intakes at different meals are independent
of one another, what is the probability that average calorie intake per day over the next 365-day year is at
most 3500? [Hint: Let X
i
, Y
i
, and Z
i
denote the three calorie intakes on day i. Then total intake is given by
X
i
Y
i
Z
i
.]
80.
The mean weight of luggage checked by a randomly selected tourist-class passenger flying between two cities on
a certain airline is 40 lb, and the standard deviation is 10 lb. The mean and standard deviation for a business-class pas-
senger are 30 lb and 6 lb, respectively. a.
If there are 12 business-class passengers and 50 tourist-class passengers on a particular flight, what are
the expected value of total luggage weight and the standard deviation of total luggage weight?
b.
If individual luggage weights are independent, normally distributed rv’s, what is the probability that total luggage
weight is at most 2500 lb?
81.
We have seen that if EX
1
EX
2
. . . EX
n
m, then EX
1
. . . X
n
nm. In some applications, the number of X
i
’s under consideration is not a fixed num- ber n but instead is an rv N. For example, let N the
number of components that are brought into a repair shop on a particular day, and let X
i
denote the repair shop time for the ith component. Then the total repair
time is X
1
X
2
. . . X
N
, the sum of a random num- ber of random variables. When N is independent of the
X
i
’s, it can be shown that E
X
1
. . . X
N
EN m
a.
If the expected number of components brought in on a particularly day is 10 and expected repair time for a ran-
domly submitted component is 40 min, what is the expected total repair time for components submitted on
any particular day?
b.
Suppose components of a certain type come in for repair according to a Poisson process with a rate of 5 per hour.
The expected number of defects per component is 3.5. What is the expected value of the total number of defects
on components submitted for repair during a 4-hour period? Be sure to indicate how your answer follows
from the general result just given.
82.
Suppose the proportion of rural voters in a certain state who favor a particular gubernatorial candidate is .45 and
the proportion of suburban and urban voters favoring the candidate is .60. If a sample of 200 rural voters and 300
urban and suburban voters is obtained, what is the approx- imate probability that at least 250 of these voters favor
this candidate?
83.
Let m denote the true pH of a chemical compound. A sequence of n independent sample pH determinations will
be made. Suppose each sample pH is a random variable
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
with expected value m and standard deviation .1. How many determinations are required if we wish the proba-
bility that the sample average is within .02 of the true pH to be at least .95? What theorem justifies your probability
calculation?
84.
If the amount of soft drink that I consume on any given day is independent of consumption on any other day and is
normally distributed with m 13 oz and s 2 and if I currently have two six-packs of 16-oz bottles, what is the
probability that I still have some soft drink left at the end of 2 weeks 14 days?
85.
Refer to Exercise 58, and suppose that the X
i
’s are inde- pendent with each one having a normal distribution. What
is the probability that the total volume shipped is at most 100,000 ft
3
?
86.
A student has a class that is supposed to end at 9:00
A
.
M
. and another that is supposed to begin at 9:10
A
.
M
. Suppose the actual ending time of the 9
A
.
M
. class is a normally distrib- uted rv X
1
with mean 9:02 and standard deviation 1.5 min and that the starting time of the next class is also a normally
distributed rv X
2
with mean 9:10 and standard deviation 1 min. Suppose also that the time necessary to get from one
classroom to the other is a normally distributed rv X
3
with mean 6 min and standard deviation 1 min. What is the prob-
ability that the student makes it to the second class before the lecture starts? Assume independence of X
1
, X
2
, and X
3
, which is reasonable if the student pays no attention to the
finishing time of the first class.
87. a.
Use the general formula for the variance of a linear com- bination to write an expression for VaX Y. Then let
a s
Y
s
X
, and show that r 1. [Hint: Variance is always 0, and CovX, Y s
X
s
Y
r .]
b.
By considering VaX Y , conclude that r 1.
c.
Use the fact that VW 0 only if W is a constant to show that r 1 only if Y aX b.
88.
Suppose a randomly chosen individual’s verbal score X and quantitative score Y on a nationally administered aptitude
examination have a joint pdf
You are asked to provide a prediction t of the individual’s total score X Y. The error of prediction is the mean
squared error E[X Y t
2
]. What value of t minimizes the error of prediction?
89. a.
Let X
1
have a chi-squared distribution with parameter n
1
see Section 4.4, and let X
2
be independent of X
1
and have a chi-squared distribution with parameter n
2
. Use the technique of Example 5.21 to show that X
1
X
2
has a chi-squared distribution with parameter n
1
n
2
. f
x, y 5 •
2 5
2x 1 3y 0 x 1, 0 y 1
otherwise
b.
In Exercise 71 of Chapter 4, you were asked to show that if Z is a standard normal rv, then Z
2
has a chi-squared distribution with n 1. Let Z
1
, Z
2
, . . . , Z
n
be n inde- pendent standard normal rv’s. What is the distribution of
? Justify your answer.
c.
Let X
1
, . . . , X
n
be a random sample from a normal dis- tribution with mean m and variance s
2
. What is the dis- tribution of the sum
? Justify your answer.
90. a.
Show that CovX, Y Z CovX, Y CovX, Z.
b.
Let X
1
and X
2
be quantitative and verbal scores on one aptitude exam, and let Y
1
and Y
2
be corresponding scores on another exam. If CovX
1
, Y
1
5, CovX
1
, Y
2
1, CovX
2
, Y
1
2, and CovX
2
, Y
2
8, what is the covariance between the two total scores X
1
X
2
and Y
1
Y
2
?
91.
A rock specimen from a particular area is randomly selected and weighed two different times. Let W denote
the actual weight and X
1
and X
2
the two measured weights. Then X
1
W E
1
and X
2
W E
2
, where E
1
and E
2
are the two measurement errors. Suppose that the E
i
s are independent of one another and of W and that .
a.
Express r, the correlation coefficient between the two measured weights X
1
and X
2
, in terms of , the variance
of actual weight, and , the variance of measured weight.
b.
Compute r when s
W
1 kg and s
E
.01 kg.
92.
Let A denote the percentage of one constituent in a ran- domly selected rock specimen, and let B denote the per-
centage of a second constituent in that same specimen. Suppose D and E are measurement errors in determining the
values of A and B so that measured values are X A D and Y B E, respectively. Assume that measurement
errors are independent of one another and of actual values. a.
Show that where X
1
and X
2
are replicate measurements on the value of A, and Y
1
and Y
2
are defined analogously with respect to B. What effect does the presence of measure-
ment error have on the correlation?
b.
What is the maximum value of CorrX, Y when CorrX
1
, X
2
.8100 and CorrY
1
, Y
2
.9025? Is this disturbing?
93.
Let X
1
, . . . , X
n
be independent rv’s with mean values m
1
, . . . , m
n
and variances . Consider a function hx
1
, . . . , x
n
, and use it to define a new rv Y hX
1
, . . . , X
n
. Under rather general conditions on the h function, if the s
i
’s are all small relative to the corresponding m
i
’s, it can be shown that E
Y ⬇ hm
1
, . . . , m
n
and
where each partial derivative is evaluated at x
1
, . . . , x
n
m
1
, . . . , m
n
. Suppose three resistors with resistances X
1
, X
2
, X
3
V Y
a h
x
1
b
2
s
1 2
1 c 1 a
h x
n
b
2
s
n 2
s
1 2
, c
, s
n 2
CorrX, Y 5 CorrA, B 1CorrX
1
, X
2
1CorrY
1
, Y
2
s
X 2
s
W 2
V E
1
5 VE
2
5 s
E 2
Y 5 g
n i5
1
[X
i
2 m s]
2
Z
1 2
1 c1 Z
n 2
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Bibliography
Devore, Jay, and Kenneth Berk, Modern Mathematical Statistics with Applications,
Thomson-BrooksCole, Belmont, CA, 2007. A bit more sophisticated exposition of probability
topics than in the present book. Olkin, Ingram, Cyrus Derman, and Leon Gleser, Probability
Models and Applications 2nd ed., Macmillan, New York,
1994. Contains a careful and comprehensive exposition of joint distributions, rules of expectation, and limit theorems.
are connected in parallel across a battery with voltage X
4
. Then by Ohm’s law, the current is
Let m
1
10 ohms, s
1
1.0 ohm, m
2
15 ohms, s
2
1.0 ohm, m
3
20 ohms, s
3
1.5 ohms, m
4
120 V, s
4
4.0 V. Calculate the approximate expected value and standard devia-
tion of the current suggested by “Random Samplings,” CHEMTECH,
1984: 696–697.
94.
A more accurate approximation to E[hX
1
, . . . , X
n
] in Exercise 93 is
h m
1
, c
,m
n
1 1
2 s
1 2
a
2
h x
1 2
b 1 c1 1
2 s
n 2
a
2
h x
n 2
b Y 5 X
4
c 1
X
1
1 1
X
2
1 1
X
3
d Compute this for Y hX
1
, X
2
, X
3
, X
4
given in Exercise 93, and compare it to the leading term hm
1
, . . . , m
n
.
95.
Let X and Y be independent standard normal random vari- ables, and define a new rv by U .6X .8Y.
a.
