Page 92 - Applied statistics and probability for engineers

70 Chapter 3/Discrete Random Variables and Probability Distributions

computers that vote for a left roll when a right roll is appropri- 3 86 17.20
ate. What is the probability mass function of X? 4 102 20.40
3-27. A disk drive manufacturer sells storage devices with 5 87 17.40
capacities of one terabyte, 500 gigabytes, and 100 gigabytes
with probabilities 0.5, 0.3, and 0.2, respectively. The revenues 6 62 12.40
associated with the sales in that year are estimated to be $50 7 40 8.00
million, $25 million, and $10 million, respectively. Let X 8 18 3.60
denote the revenue of storage devices during that year. Deter- 9 14 2.80
mine the probability mass function of X. 10 11 2.20
3-28. Marketing estimates that a new instrument for the 15 10 2.00
analysis of soil samples will be very successful, moderately
successful, or unsuccessful with probabilities 0.3, 0.6, and Calculate the probability mass function of the wait time for
0.1, respectively. The yearly revenue associated with a very service.
successful, moderately successful, or unsuccessful product is 3-34. The distribution of the time until a Web site changes is
$10 million, $5 million, and $1 million, respectively. Let the important to Web crawlers that search engines use to maintain
random variable X denote the yearly revenue of the product. current information about Web sites. The distribution of the
Determine the probability mass function of X. time until change (in days) of a Web site is approximated in
3-29. The distributor of a machine for cytogenics has the following table.
developed a new model. The company estimates that when it Days until Probability
is introduced into the market, it will be very successful with a Changes
probability 0.6, moderately successful with a probability 0.3, 1.5 0.05
and not successful with probability 0.1. The estimated yearly 3.0 0.25
proit associated with the model being very successful is $15
million and with it being moderately successful is $5 million; 4.5 0.35
not successful would result in a loss of $500,000. Let X be the 5.0 0.20
yearly proit of the new model. Determine the probability mass 7.0 0.15
function of X. Calculate the probability mass function of the days until change.
3-30. An assembly consists of two mechanical compo- 3-35 The following table shows the typical depth (rounded to
nents. Suppose that the probabilities that the irst and second the nearest foot) for nonfailed wells in geological formations
components meet speciications are 0.95 and 0.98, respectively. in Baltimore County (The Journal of Data Science, 2009, Vol.
Assume that the components are independent. Determine the 7, pp. 111–127).
probability mass function of the number of components in the
assembly that meet speciications.
3-31. An assembly consists of three mechanical com- Geological Number of Nonfailed
ponents. Suppose that the probabilities that the irst, second, Formation Group Nonfailed Wells Well Depth
and third components meet speciications are 0.95, 0.98, and Gneiss 1,515 255
0.99, respectively. Assume that the components are independ- Granite 26 218
ent. Determine the probability mass function of the number of Loch Raven Schist 3,290 317
components in the assembly that meet speciications.
3-32. The data from 200 endothermic reactions involving Maic 349 231
sodium bicarbonate are summarized as follows: Marble 280 267
Prettyboy Schist 1,343 255
Final Temperature Number of Other schists 887 267
Conditions Reactions
Serpentine 36 217
266 K 48
Total 7,726 2,027
271 K 60
274 K 92 Calculate the probability mass function of depth for nonfailed
wells from the table.
Calculate the probability mass function of inal temperature. 3-36. Consider the wafers with contamination particles in
3-33. Actual lengths of stay at a hospital’s emergency depart-
Example 2-17. Assume that wafers are independent with
ment in 2009 are shown in the following table (rounded to the
respect to contamination particles. Wafers are selected until
nearest hour). Length of stay is the total of wait and service times.
one with ive or more contamination particles occurs. What
Some longer stays are also approximated as 15 hours in this table.
is the probability mass function of the number of wafers
Hours Count Percent selected?
3-37. Consider the circuit in Example 2-32. Assume that
1 19 3.80 devices fail independently. What is the probability mass func-
2 51 10.20 tion of the number of failed devices?