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5-5 COVARIANCE AND CORRELATION 171

EXERCISES FOR SECTION 5–4
5-55. Suppose the random variables X, Y, and Z have the joint (c) Conditional probability distribution of X given that Y
probability density function f 1x, y, z2 8xyz for 0 x 1, 0.5 and Z 0.5
0 y 1, and 0 z 1. Determine the following: (d) Conditional probability distribution of X given that
(a) P1X 0.52 (b) P1X 0.5, Y 0.52 Y 0.5
(c) P1Z 22 (d) P1X 0.5 or Z 22 5-64. The yield in pounds from a day’s production is nor-
(e) E1X 2 mally distributed with a mean of 1500 pounds and standard
5-56. Continuation of Exercise 5-55. Determine the following: deviation of 100 pounds. Assume that the yields on different
(a) P1X 0.5 ƒ Y 0.52 days are independent random variables.
(b) P1X 0.5, Y 0.5 ƒ Z 0.82 (a) What is the probability that the production yield exceeds
1400 pounds on each of five days next week?
5-57. Continuation of Exercise 5-55. Determine the following:
(a) Conditional probability distribution of X given that Y (b) What is the probability that the production yield exceeds
0.5 and Z 0.8 1400 pounds on at least four of the five days next week?
(b) P1X 0.5 ƒ Y 0.5, Z 0.82 5-65. The weights of adobe bricks used for construction are
normally distributed with a mean of 3 pounds and a standard
5-58. Suppose the random variables X, Y, and Z have
the joint probability density function f XYZ (x, y, z) c over deviation of 0.25 pound. Assume that the weights of the bricks
2
2
the cylinder x y 4 and 0 z 4. Determine the are independent and that a random sample of 20 bricks is
following. selected.
(a) The constant c so that f XYZ (x, y, z) is a probability density (a) What is the probability that all the bricks in the sample
function exceed 2.75 pounds?
2
2
(b) P1X Y 22 (b) What is the probability that the heaviest brick in the sam-
ple exceeds 3.75 pounds?
(c) P1Z 22
5-66. A manufacturer of electroluminescent lamps knows
(d) E1X 2
that the amount of luminescent ink deposited on one of
5-59. Continuation of Exercise 5-58. Determine the
its products is normally distributed with a mean of 1.2
following: grams and a standard deviation of 0.03 grams. Any lamp
2
2
(a) P1X 1 ƒ Y 12 (b) P1X Y 1 ƒ Z 12
with less than 1.14 grams of luminescent ink will fail
5-60. Continuation of Exercise 5-58. Determine the condi- to meet customer’s specifications. A random sample of
tional probability distribution of Z given that X 1 and 25 lamps is collected and the mass of luminescent ink on
Y 1. each is measured.
5-61. Determine the value of c that makes f XYZ (x, y, z) c (a) What is the probability that at least 1 lamp fails to meet
a joint probability density function over the region x
0, specifications?
y
0, z
0, and x y z 1. (b) What is the probability that 5 lamps or fewer fail to meet
5-62. Continuation of Exercise 5-61. Determine the following: specifications?
(a) P1X 0.5, Y 0.5, Z 0.52 (c) What is the probability that all lamps conform to specifi-
(b) P1X 0.5, Y 0.52 cations?
(c) P1X 0.52 (d) Why is the joint probability distribution of the 25 lamps
(d) E1X 2 not needed to answer the previous questions?
5-63. Continuation of Exercise 5-61. Determine the following:
(a) Marginal distribution of X
(b) Joint distribution of X and Y

5-5 COVARIANCE AND CORRELATION

When two or more random variables are defined on a probability space, it is useful to describe
how they vary together; that is, it is useful to measure the relationship between the variables.
A common measure of the relationship between two random variables is the covariance. To
define the covariance, we need to describe the expected value of a function of two random
variables h(X, Y). The definition simply extends that used for a function of a single random
variable.