Page 85 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 63

3-3 CUMULATIVE DISTRIBUTION FUNCTIONS 63

EXERCISES FOR SECTION 3-2
3-13. The sample space of a random experiment is {a, b, c, probability 0.5, a storage device with 500 gigabytes capacity
d, e, f}, and each outcome is equally likely. A random variable will sell with a probability 0.3, and a storage device with 100
is defined as follows: gigabytes capacity will sell with probability 0.2. The revenue
associated with the sales in that year are estimated to be $50
outcome a b c d e f million, $25 million, and $10 million, respectively. Let X be
the revenue of storage devices during that year. Determine the
x 0 0 1.5 1.5 2 3
probability mass function of X.
Determine the probability mass function of X. 3-21. An optical inspection system is to distinguish
3-14. Use the probability mass function in Exercise 3-11 to among different part types. The probability of a correct
determine the following probabilities: classification of any part is 0.98. Suppose that three parts
are inspected and that the classifications are independent.
(a) P1X 1.52 (b) P10.5 X 2.72
(c) P1X 32 (d) P10 X 22 Let the random variable X denote the number of parts that
are correctly classified. Determine the probability mass
(e) P1X 0 or X 22 function of X.
Verify that the following functions are probability mass func-
tions, and determine the requested probabilities. 3-22. In a semiconductor manufacturing process, three
wafers from a lot are tested. Each wafer is classified as pass or
3-15. x 2 1 0 1 2 fail. Assume that the probability that a wafer passes the test is
0.8 and that wafers are independent. Determine the probabil-
f 1x2 1 8 2 8 2 8 2 8 1 8 ity mass function of the number of wafers from a lot that pass
the test.
(a) P1X 22 (b) P1X 22
(c) P1 1 X 12 (d) P1X 1 or X 22 3-23. The distributor of a machine for cytogenics has
x
3-16. f 1x2 18 7211 22 , x 1, 2, 3 developed a new model. The company estimates that when it
is introduced into the market, it will be very successful with a
(a) P1X 12 (b) P1X 12
probability 0.6, moderately successful with a probability 0.3,
(c) P12 X 62 (d) P1X 1 or X 12
and not successful with probability 0.1. The estimated yearly
2x 1 profit associated with the model being very successful is $15
3-17. f 1x2 , x 0, 1, 2, 3, 4 million and being moderately successful is $5 million; not
25
(a) P1X 42 (b) P1X 12 successful would result in a loss of $500,000. Let X be the
yearly profit of the new model. Determine the probability
(c) P12 X 42 (d) P1X 102 mass function of X.
x
3-18. f 1x2 13 4211 42 , x 0, 1, 2, p
3-24. An assembly consists of two mechanical components.
(a) P1X 22 (b) P1X 22
Suppose that the probabilities that the first and second compo-
(c) P1X 22 (d) P1X 12 nents meet specifications are 0.95 and 0.98. Assume that the
3-19. Marketing estimates that a new instrument for the components are independent. Determine the probability mass
analysis of soil samples will be very successful, moderately function of the number of components in the assembly that
successful, or unsuccessful, with probabilities 0.3, 0.6, meet specifications.
and 0.1, respectively. The yearly revenue associated with
a very successful, moderately successful, or unsuccessful 3-25. An assembly consists of three mechanical compo-
product is $10 million, $5 million, and $1 million, respec- nents. Suppose that the probabilities that the first, second, and
tively. Let the random variable X denote the yearly revenue of third components meet specifications are 0.95, 0.98, and 0.99.
the product. Determine the probability mass function of X. Assume that the components are independent. Determine the
probability mass function of the number of components in the
3-20. A disk drive manufacturer estimates that in five years assembly that meet specifications.
a storage device with 1 terabyte of capacity will sell with

3-3 CUMULATIVE DISTRIBUTION FUNCTIONS

EXAMPLE 3-6 In Example 3-4, we might be interested in the probability of three or fewer bits being in error.
This question can be expressed as P1X 32.
The event that 5X 36 is the union of the events 5X 06, 5X 16, 5X 26, and