5-5 COVARIANCE AND CORRELATION 171EXERCISES FOR SECTION 5–4
5-55. Suppose the random variables X, Y, and Zhave the joint
probability density function for 0 x 1,
0 y 1, and 0 z 1. Determine the following:
(a) (b)
(c) (d)
(e)
5-56. Continuation of Exercise 5-55. Determine the following:
(a)
(b)
5-57. Continuation of Exercise 5-55. Determine the following:
(a) Conditional probability distribution of Xgiven that Y
0.5 and Z0.8
(b)
5-58. Suppose the random variables X, Y, and Zhave
the joint probability density function fXYZ(x,y,z)cover
the cylinder x^2 y^2 4 and 0z4. Determine the
following.
(a) The constant cso that fXYZ(x,y,z) is a probability density
function
(b)
(c)
(d)
5-59. Continuation of Exercise 5-58. Determine the
following:
(a) (b)
5-60. Continuation of Exercise 5-58. Determine the condi-
tional probability distribution of Zgiven that X1 and
Y1.
5-61. Determine the value of cthat makes fXYZ(x,y,z)c
a joint probability density function over the region x 0,
y 0,z 0, and xyz 1.
5-62. Continuation of Exercise 5-61. Determine the following:
(a)
(b)
(c)
(d)
5-63. Continuation of Exercise 5-61. Determine the following:
(a) Marginal distribution of X
(b) Joint distribution of Xand YE 1 X 2P 1 X0.5 2P 1 X0.5, Y0.5 2P 1 X0.5, Y0.5, Z0.5 2P 1 X 1 ƒY 12 P 1 X^2 Y^2 1 ƒZ 12E 1 X 2P 1 Z 22P 1 X^2 Y^2 22P 1 X0.5ƒY0.5, Z0.8 2P 1 X0.5, Y0.5ƒZ0.8 2P 1 X0.5ƒY0.5 2E 1 X 2P 1 Z 22 P 1 X0.5 or Z 22P 1 X0.5 2 P 1 X0.5, Y0.5 2f 1 x, y, z 2 8 xyz5-5 COVARIANCE AND CORRELATIONWhen 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.(c) Conditional probability distribution of Xgiven that Y
0.5 and Z0.5
(d) Conditional probability distribution of X given that
Y0.5
5-64. The yield in pounds from a day’s production is nor-
mally distributed with a mean of 1500 pounds and standard
deviation of 100 pounds. Assume that the yields on different
days are independent random variables.
(a) What is the probability that the production yield exceeds
1400 pounds on each of five days next week?
(b) What is the probability that the production yield exceeds
1400 pounds on at least four of the five days next week?
5-65. The weights of adobe bricks used for construction are
normally distributed with a mean of 3 pounds and a standard
deviation of 0.25 pound. Assume that the weights of the bricks
are independent and that a random sample of 20 bricks is
selected.
(a) What is the probability that all the bricks in the sample
exceed 2.75 pounds?
(b) What is the probability that the heaviest brick in the sam-
ple exceeds 3.75 pounds?
5-66. A manufacturer of electroluminescent lamps knows
that the amount of luminescent ink deposited on one of
its products is normally distributed with a mean of 1.2
grams and a standard deviation of 0.03 grams. Any lamp
with less than 1.14 grams of luminescent ink will fail
to meet customer’s specifications. A random sample of
25 lamps is collected and the mass of luminescent ink on
each is measured.
(a) What is the probability that at least 1 lamp fails to meet
specifications?
(b) What is the probability that 5 lamps or fewer fail to meet
specifications?
(c) What is the probability that all lamps conform to specifi-
cations?
(d) Why is the joint probability distribution of the 25 lamps
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