Page 114 - Applied statistics and probability for engineers

92 Chapter 3/Discrete Random Variables and Probability Distributions

What are the mean and standard deviation of the number of (d) What is the mean number of reactions until two reactions
attempts before the hacker selects a user password? result in inal temperatures less than 272 K?
(c) Comment on the security differences between six- and 3-135. A Web site randomly selects among 10 products to discount
three-character passwords. each day. The color printer of interest to you is discounted today.
3-129. A trading company uses eight computers to trade on (a) What is the expected number of days until this product is
the New York Stock Exchange (NYSE). The probability of a again discounted?
computer failing in a day is 0.005, and the computers fail inde- (b) What is the probability that this product is irst discounted
pendently. Computers are repaired in the evening, and each day again exactly 10 days from now?
is an independent trial. (c) If the product is not discounted for the next ive days, what
(a) What is the probability that all eight computers fail in a day? is the probability that it is irst discounted again 15 days
(b) What is the mean number of days until a speciic computer fails? from now?
(c) What is the mean number of days until all eight computers (d) What is the probability that this product is irst discounted
fail on the same day? again within three or fewer days?
3-130. Assume that 20 parts are checked each hour and that X 3-136. Consider the visits that result in leave without being
denotes the number of parts in the sample of 20 that require rework. seen (LWBS) at an emergency department in Example 2-8.
Parts are assumed to be independent with respect to rework. Assume that people independently arrive for service at hospital l.
(a) If the percentage of parts that require rework remains at (a) What is the probability that the ifth visit is the irst one to
1%, what is the probability that hour 10 is the irst sample LWBS?
at which X exceeds 1? (b) What is the probability that either the ifth or sixth visit is
(b) If the rework percentage increases to 4%, what is the proba- the irst one to LWBS?
bility that hour 10 is the irst sample at which X exceeds 1? (c) What is the probability that the irst visit to LWBS is among
(c) If the rework percentage increases to 4%, what is the the irst four visits?
expected number of hours until X exceeds 1? (d) What is the expected number of visits until the third LWBS
3-131. A fault-tolerant system that processes transactions for occurs?
a inancial services irm uses three separate computers. If the 3-137. Consider the time to recharge the lash in cell-phone
operating computer fails, one of the two spares can be immedi- cameras as in Example 3-2. Assume that the probability that
ately switched online. After the second computer fails, the last a camera passes the test is 0.8 and the cameras perform inde-
computer can be immediately switched online. Assume that the pendently. Determine the following:
−
8
probability of a failure during any transaction is 10 and that (a) Probability that the second failure occurs on the tenth cam-
the transactions can be considered to be independent events. era tested.
(a) What is the mean number of transactions before all comput- (b) Probability that the second failure occurs in tests of four or
ers have failed? fewer cameras.
(b) What is the variance of the number of transactions before (b) Expected number of cameras tested to obtain the third
all computers have failed? failure.
3-132. In the process of meiosis, a single parent diploid 3-138. An array of 30 LED bulbs is used in an automotive
cell goes through eight different phases. However, only 60% light. The probability that a bulb is defective is 0.001 and
of the processes pass the irst six phases and only 40% pass all defective bulbs occur independently. Determine the following:
eight. Assume that the results from each phase are independent. (a) Probability that an automotive light has two or more defec-
(a) If the probability of a successful pass of each one of the irst tive bulbs.
six phases is constant, what is the probability of a success- (b) Expected number of automotive lights to check to obtain
ful pass of a single one of these phases? one with two or more defective bulbs.
(b) If the probability of a successful pass of each one of the last 3-139. Consider the patient data in Example 2-8. Suppose that
two phases is constant, what is the probability of a success- patients are randomly selected with replacement, from the total
ful pass of a single one of these phases? for hospital 4. Determine the following:
3-133. Show that the probability density function of a negative (a) Probability that the irst patient admitted is the irst one
binomial random variable equals the probability density function of selected.
a geometric random variable when r = 1. Show that the formulas for (b) Probability that four or fewer patients are selected to
the mean and variance of a negative binomial random variable equal admit two.
the corresponding results for a geometric random variable when r = 1. (c) Expected number of patients selected to admit 10.
3-134. Consider the endothermic reactions in Exercise 3-32. 3-140. Customers visit a Web site, and the probability of an
Assume that independent reactions are conducted. order if a customer views ive or fewer pages is 0.01. However,
(a) What is the probability that the irst reaction to result in a if a customer views more than ive pages, the probability of
inal temperature less than 272 K is the tenth reaction? an order is 0.1. The probability a customer views ive or more
(b) What is the mean number of reactions until the irst inal pages is 0.25. The customers behave independently.
temperature is less than 272 K? (a) Is the number of customers who visit the site until an order
(c) What is the probability that the irst reaction to result in a is obtained a geometric random variable? Why or why not?
inal temperature less than 272 K occurs within three or (b) What is the probability that the irst order is obtained from
fewer reactions? the tenth customer to visit the site?