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.