18 Let X and Y be discrete rv’s with joint pmf
Example 5.18 Let X and Y be discrete rv’s with joint pmf
p(x, y) 5 u 4
(x, y) 5 (24, 1), (4,21), (2, 2), (22, 22)
0 otherwise
5.2 Expected Values, Covariance, and Correlation
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
0 and E(XY) 5 (24) 1 4 1 1 (24) 1 4 1 1 (4) 4 1 (4) 4 50 . The covariance is then
are completely dependent. However, by symmetry m X m Y
Cov(X,Y) E(XY) 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!
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.
24. Six individuals, including A and B, take seats around a cir-
For a randomly selected student, let X the number of
cular table in a completely random fashion. Suppose the
points earned on the first part and Y the number of points
seats are numbered 1, . . . , 6. Let X A’s seat number and
earned on the second part. Suppose that the joint pmf of
Y B’s seat number. If A sends a written message around
X and Y is given in the accompanying table.
the table to B in the direction in which they are closest, how many individuals (including A and B) would you expect to
y
handle the message?
p(x, y)
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.
a. If the score recorded in the grade book is the total num-
Suppose that X and Y are independent and that each is uniformly
ber of points earned on the two parts, what is the
distributed on the interval [L
A, L A] (where 0 A L).
expected recorded score E(X
Y)? What is the expected area of the resulting rectangle?
b. If the maximum of the two scores is recorded, what is the
26. Consider a small ferry that can accommodate cars and
expected recorded score?
buses. The toll for cars is 3, and the toll for buses is 10.
23. The difference between the number of customers in line at
Let X and Y denote the number of cars and buses, respec-
the express checkout and the number in line at the super-
tively, carried on a single trip. Suppose the joint distribution
express checkout in Exercise 3 is X 1 X . Calculate the
of X and Y is as given in the table of Exercise 7. Compute
expected difference.
the expected revenue from a single trip.
CHAPTER 5 Joint Probability Distributions and Random Samples
27. Annie and Alvie have agreed to meet for lunch between
32. Reconsider the minicomputer component lifetimes X and Y
noon (0:00 P . M .) and 1:00 P . M . Denote Annie’s arrival time
as described in Exercise 12. Determine E(XY). What can be
by X, Alvie’s by Y, and suppose X and Y are independent
said about Cov(X, Y) and r?
with pdf’s
33. Use the result of Exercise 28 to show that when X and Y are
3x 2 0x1
independent, Cov(X, Y) Corr(X, Y) 0.
f X (x) 5 e
0 otherwise
34. a. Recalling the definition of s 2 for a single rv X, write a
2y 0 y 1
formula that would be appropriate for computing the f Y ( y) 5 e variance of a function h(X, Y) of two random variables.
0 otherwise
[Hint: Remember that variance is just a special expected
What is the expected amount of time that the one who
value.]
arrives first must wait for the other person? [Hint: h(X, Y )
b. Use this formula to compute the variance of the
recorded score h(X, Y) [ max(X, Y)] in part (b) of
28. Show that if X and Y are independent rv’s, then E(XY )
35. a. Use the rules of expected value to show that Cov(aX
E(X)
E(Y). Then apply this in Exercise 25. [Hint: Consider the continuous case with f(x, y)
f b, cY
d) ac Cov(X, Y).
X (x)
Y f ( y).]
b. Use part (a) along with the rules of variance and standard
29. Compute the correlation coefficient r for X and Y of
deviation to show that Corr(aX
b, cY d) Corr(X,
Example 5.16 (the covariance has already been
Y) when a and c have the same sign.
computed).
c. What happens if a and c have opposite signs?
30. a. Compute the covariance for X and Y in Exercise 22.
36. Show that if Y aX b (a ⬆ 0), then Corr(X, Y) 1 or
b. Compute r for X and Y in the same exercise.
1. Under what conditions will r 1?
31. a. Compute the covariance between X and Y in Exercise 9.
b. Compute the correlation coefficient r for this X and Y.