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156 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

A sample of 20 slabs is selected for testing. Let X, Y, and Z 5-28. Continuation of Exercise 5-27. Determine the
denote the number of slabs that are independently classified as following:
high, medium, and low, respectively. (a) The joint probability mass function of the number of ovens
(a) What is the name and the values of the parameters of the with a major defect and the number with a minor defect.
joint probability distribution of X, Y, and Z? (b) The expected number of ovens with a major defect.
(b) What is the range of the joint probability distribution of (c) The expected number of ovens with a minor defect.
X, Y, Z? 5-29. Continuation of Exercise 5-27. Determine the follow-
(c) What is the name and the values of the parameters of the ing:
marginal probability distribution of X? (a) The conditional probability that two ovens have major
(d) Determine E1X 2 and V 1X 2 . defects given that two ovens have minor defects
5-21. Continuation of Exercise 5-20. Determine the (b) The conditional probability that three ovens have major
following: defects given that two ovens have minor defects
(a) P 1X 1, Y 17, Z 32 (c) The conditional probability distribution of the number of
(b) P 1X 1, Y 17, Z 32 ovens with major defects given that two ovens have minor
(c) P 1X 12 defects
(d) E 1X2 (d) The conditional mean of the number of ovens with major
5-22. Continuation of Exercise 5-20. Determine the defects given that two ovens have minor defects
following: 5-30. In the transmission of digital information, the proba-
(a) P 1X 2, Z 3 ƒ Y 172 (b) P 1X 2 ƒ Y 172 bility that a bit has high, moderate, or low distortion is 0.01,
(c) E 1X 0 Y 172 0.04, and 0.95, respectively. Suppose that three bits are trans-
5-23. An order of 15 printers contains four with a graphics- mitted and that the amount of distortion of each bit is assumed
enhancement feature, five with extra memory, and six with to be independent.
both features. Four printers are selected at random, without (a) What is the probability that two bits have high distortion
replacement, from this set. Let the random variables X, Y, and one has moderate distortion?
and Z denote the number of printers in the sample (b) What is the probability that all three bits have low
with graphics enhancement only, extra memory only, and distortion?
both, respectively. 5-31. Continuation of Exercise 5-30. Let X and Y denote the
(a) Describe the range of the joint probability distribution of number of bits with high and moderate distortion out of the
X, Y, and Z. three transmitted, respectively. Determine the following:
(b) Is the probability distribution of X, Y, and Z multinomial? (a) The probability distribution, mean and variance of X.
Why or why not? (b) The conditional probability distribution, conditional mean
5-24. Continuation of Exercise 5-23. Determine the condi- and conditional variance of X given that Y 2.
tional probability distribution of X given that Y 2. 5-32. A marketing company performed a risk analysis for a
5-25. Continuation of Exercise 5-23. Determine the follow- manufacturer of synthetic fibers and concluded that new com-
ing: petitors present no risk 13% of the time (due mostly to the di-
(a) P1X 1, Y 2, Z 12 (b) P1X 1, Y 12 versity of fibers manufactured), moderate risk 72% of the time
(c) E1X 2 and V1X 2 (some overlapping of products), and very high risk (competi-
tor manufactures the exact same products) 15% of the time. It
5-26. Continuation of Exercise 5-23. Determine the
is known that 12 international companies are planning to open
following:
new facilities for the manufacture of synthetic fibers within
(a) P1X 1, Y 2 ƒ Z 12 (b) P1X 2 ƒ Y 22
the next three years. Assume the companies are independent.
(c) The conditional probability distribution of X given that
Let X, Y, and Z denote the number of new competitors that will
Y 0 and Z 3.
pose no, moderate, and very high risk for the interested com-
5-27. Four electronic ovens that were dropped during ship- pany, respectively.
ment are inspected and classified as containing either a major, (a) What is the range of the joint probability distribution of
a minor, or no defect. In the past, 60% of dropped ovens had X, Y, and Z?
a major defect, 30% had a minor defect, and 10% had no (b) Determine P(X 1, Y 3, Z 1)
defect. Assume that the defects on the four ovens occur (c) Determine P1Z 22
independently. 5-33. Continuation of Exercise 5-32. Determine the
(a) Is the probability distribution of the count of ovens in each following:
category multinomial? Why or why not? (a) P 1Z 2 ƒ Y 1, X 102 (b) P 1Z 1 ƒ X 102
(b) What is the probability that, of the four dropped ovens, two
have a major defect and two have a minor defect? (c) P 1Y 1, Z 1 ƒ X 102 (d) E 1Z ƒ X 102
(c) What is the probability that no oven has a defect?