The points that receive positive probability mass are identified on the x, y coordinate system in Figure 5.5. It is evident from the figure that the value of X
is completely determined by the value of Y and vice versa, so the two variables are completely dependent. However, by symmetry m
X
m
Y
0 and . The covariance is then
E XY 5 24
1 4
1 24
1 4
1 4
1 4
1 4
1 4
5
2 1
ⴚ1 ⴚ2
1 2
3 4
ⴚ1 ⴚ2
ⴚ3 ⴚ4
Figure 5.5 The population of pairs for Example 5.18
■ A value of r near 1 does not necessarily imply that increasing the value of X causes
Y to increase. It implies only that large X values are associated with large Y values.
For example, in the population of children, vocabulary size and number of cavities are quite positively correlated, but it is certainly not true that cavities cause vocabu-
lary to grow. Instead, the values of both these variables tend to increase as the value of age, a third variable, increases. For children of a fixed age, there is probably a low
correlation between number of cavities and vocabulary size. In summary, association a high correlation is not the same as causation.
EXERCISES
Section 5.2 22–36
22.
An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X the number of
points earned on the first part and Y the number of points earned on the second part. Suppose that the joint pmf of
X
and Y is given in the accompanying table. y
p x, y
5 10
15 .02
.06 .02
.10 x
5 .04
.15 .20
.10 10
.01 .15
.14 .01
a.
If the score recorded in the grade book is the total num- ber of points earned on the two parts, what is the
expected recorded score EX Y?
b.
If the maximum of the two scores is recorded, what is the expected recorded score?
23.
The difference between the number of customers in line at the express checkout and the number in line at the super-
express checkout in Exercise 3 is X
1
X
2
. Calculate the expected difference.
24.
Six individuals, including A and B, take seats around a cir- cular table in a completely random fashion. Suppose the
seats are numbered 1, . . . , 6. Let X A’s seat number and Y
B’s seat number. If A sends a written message around the table to B in the direction in which they are closest, how
many individuals including A and B would you expect to handle the message?
25.
A surveyor wishes to lay out a square region with each side hav- ing length L. However, because of a measurement error, he
instead lays out a rectangle in which the north–south sides both have length X and the east–west sides both have length Y.
Suppose that X and Y are independent and that each is uniformly distributed on the interval [L A, L A] where 0
A
L.
What is the expected area of the resulting rectangle?
26.
Consider a small ferry that can accommodate cars and buses. The toll for cars is 3, and the toll for buses is 10.
Let X and Y denote the number of cars and buses, respec- tively, carried on a single trip. Suppose the joint distribution
of X and Y is as given in the table of Exercise 7. Compute the expected revenue from a single trip.
CovX,Y EXY m
X
m
Y
0 and thus r
X,Y
0. Although there is perfect dependence, there is also complete absence of any linear relationship
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 5.19
5.3
Statistics and Their Distributions
The observations in a single sample were denoted in Chapter 1 by x
1
, x
2
, . . . , x
n
. Consider selecting two different samples of size n from the same population dis-
tribution. The x
i
’s in the second sample will virtually always differ at least a bit from those in the first sample. For example, a first sample of n 3 cars of a par-
ticular type might result in fuel efficiencies x
1
30.7, x
2
29.4, x
3
31.1, whereas a second sample may give x
1
28.8, x
2
30.0, and x
3
32.5. Before we obtain data, there is uncertainty about the value of each x
i
. Because of this uncertainty, before the data becomes available we view each observation as a ran-
dom variable and denote the sample by X
1
, X
2
, . . . , X
n
uppercase letters for random variables.
This variation in observed values in turn implies that the value of any func- tion of the sample observations—such as the sample mean, sample standard devi-
ation, or sample fourth spread—also varies from sample to sample. That is, prior to obtaining x
1
, . . . , x
n
, there is uncertainty as to the value of , the value of s, and so on.
Suppose that material strength for a randomly selected specimen of a particular type has a Weibull distribution with parameter values a 2 shape and b 5
scale. The corresponding density curve is shown in Figure 5.6. Formulas from Section 4.5 give
The mean exceeds the median because of the distribution’s positive skew. m 5 E
x 5 4.4311 m | 5 4.1628 s
2
5 V X 5 5.365 s 5 2.316
x
27.
Annie and Alvie have agreed to meet for lunch between noon 0:00
P
.
M
. and 1:00
P
.
M
. Denote Annie’s arrival time by X, Alvie’s by Y, and suppose X and Y are independent
with pdf’s
What is the expected amount of time that the one who arrives first must wait for the other person? [Hint: hX, Y
| X Y | .]
28.
Show that if X and Y are independent rv’s, then EXY E
X EY. Then apply this in Exercise 25. [Hint: Consider the continuous case with fx, y f
X
x f
Y
y.]
29.
Compute the correlation coefficient r for X and Y of Example 5.16 the covariance has already been
computed.
30. a.
Compute the covariance for X and Y in Exercise 22.
b.
Compute r for X and Y in the same exercise.
31. a.
Compute the covariance between X and Y in Exercise 9.
b.
Compute the correlation coefficient r for this X and Y. f
Y
y 5 e
2y 0 y 1 0 otherwise
f
X
x 5 e
3x
2
0 x 1 otherwise
32.
Reconsider the minicomputer component lifetimes X and Y as described in Exercise 12. Determine EXY. What can be
said about CovX, Y and r?
33.
Use the result of Exercise 28 to show that when X and Y are independent, CovX, Y CorrX, Y 0.
34. a.
Recalling the definition of s
2
for a single rv X, write a formula that would be appropriate for computing the
variance of a function hX, Y of two random variables. [Hint: Remember that variance is just a special expected
value.]
b.
Use this formula to compute the variance of the recorded score hX, Y [ maxX, Y] in part b of
Exercise 22.
35. a.
Use the rules of expected value to show that CovaX b
, cY d ac CovX, Y.
b.
Use part a along with the rules of variance and standard deviation to show that CorraX b, cY d CorrX,
Y when a and c have the same sign.
c.
What happens if a and c have opposite signs?
36.
Show that if Y aX b a ⬆ 0, then CorrX, Y 1 or 1. Under what conditions will r 1?
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.
5 10
15 .05
.10 .15
x f
x
Figure 5.6 The Weibull density curve for Example 5.19
We used statistical software to generate six different samples, each with n 10, from this distribution material strengths for six different groups of ten specimens each. The
results appear in Table 5.1, followed by the values of the sample mean, sample median, and sample standard deviation for each sample. Notice first that the ten observations
in any particular sample are all different from those in any other sample. Second, the six values of the sample mean are all different from one another, as are the six values
of the sample median and the six values of the sample standard deviation. The same is true of the sample 10 trimmed means, sample fourth spreads, and so on.
Table 5.1 Samples from the Weibull Distribution of Example 5.19
Sample 1
2 3
4 5
6
1 6.1171
5.07611 3.46710
1.55601 3.12372
8.93795 2
4.1600 6.79279
2.71938 4.56941
6.09685 3.92487
3 3.1950
4.43259 5.88129
4.79870 3.41181
8.76202 4
0.6694 8.55752
5.14915 2.49759
1.65409 7.05569
5 1.8552
6.82487 4.99635
2.33267 2.29512
2.30932 6
5.2316 7.39958
5.86887 4.01295
2.12583 5.94195
7 2.7609
2.14755 6.05918
9.08845 3.20938
6.74166 8
10.2185 8.50628
1.80119 3.25728
3.23209 1.75468
9 5.2438
5.49510 4.21994
3.70132 6.84426
4.91827 10
4.5590 4.04525
2.12934 5.50134
4.20694 7.26081
4.401 5.928
4.229 4.132
3.620 5.761
4.360 6.144
4.608 3.857
3.221 6.342
s 2.642
2.062 1.611
2.124 1.678
2.496 x|
x
Furthermore, the value of the sample mean from any particular sample can be regarded as a point estimate “point” because it is a single number, corresponding to
a single point on the number line of the population mean m, whose value is known to be 4.4311. None of the estimates from these six samples is identical to what is
being estimated. The estimates from the second and sixth samples are much too large, whereas the fifth sample gives a substantial underestimate. Similarly, the sam-
ple standard deviation gives a point estimate of the population standard deviation. All six of the resulting estimates are in error by at least a small amount.
In summary, the values of the individual sample observations vary from sample to sample, so will in general the value of any quantity computed from sample data, and
the value of a sample characteristic used as an estimate of the corresponding popula- tion characteristic will virtually never coincide with what is being estimated.
■
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.
