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3-7 GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS 83

3-73. The probability of a successful optical alignment in (c) If the rework percentage increases to 4%, what is the
the assembly of an optical data storage product is 0.8. Assume expected number of hours until X exceeds 1?
the trials are independent. 3-79. Consider a sequence of independent Bernoulli trials
(a) What is the probability that the first successful alignment with p 0.2.
requires exactly four trials? (a) What is the expected number of trials to obtain the first
(b) What is the probability that the first successful alignment success?
requires at most four trials? (b) After the eighth success occurs, what is the expected num-
(c) What is the probability that the first successful alignment ber of trials to obtain the ninth success?
requires at least four trials?
3-80. Show that the probability density function of a nega-
3-74. In a clinical study, volunteers are tested for a gene tive binomial random variable equals the probability density
that has been found to increase the risk for a disease. The function of a geometric random variable when r 1. Show
probability that a person carries the gene is 0.1. that the formulas for the mean and variance of a negative bi-
(a) What is the probability 4 or more people will have to be nomial random variable equal the corresponding results for
tested before 2 with the gene are detected? geometric random variable when r 1.
(b) How many people are expected to be tested before 2 with
3-81. Suppose that X is a negative binomial random variable
the gene are detected?
with p 0.2 and r 4. Determine the following:
3-75. Assume that each of your calls to a popular radio station (a) E1X 2 (b) P1X 202
has a probability of 0.02 of connecting, that is, of not obtaining a (c) P1X 192 (d) P1X 212
busy signal. Assume that your calls are independent. (e) The most likely value for X
(a) What is the probability that your first call that connects is 3-82. The probability is 0.6 that a calibration of a transducer
your tenth call? in an electronic instrument conforms to specifications for the
(b) What is the probability that it requires more than five calls measurement system. Assume the calibration attempts are
for you to connect? independent. What is the probability that at most three
(c) What is the mean number of calls needed to connect?
calibration attempts are required to meet the specifications for
3-76. In Exercise 3-70, recall that a particularly long traffic the measurement system?
light on your morning commute is green 20% of the time that
3-83. An electronic scale in an automated filling operation
you approach it. Assume that each morning represents an
stops the manufacturing line after three underweight packages
independent trial.
are detected. Suppose that the probability of an underweight
(a) What is the probability that the first morning that the light
package is 0.001 and each fill is independent.
is green is the fourth morning that you approach it?
(a) What is the mean number of fills before the line is
(b) What is the probability that the light is not green for 10
stopped?
consecutive mornings?
(b) What is the standard deviation of the number of fills
3-77. A trading company has eight computers that it uses to before the line is stopped?
trade on the New York Stock Exchange (NYSE). The proba-
3-84. A fault-tolerant system that processes transactions for
bility of a computer failing in a day is 0.005, and the comput-
a financial services firm uses three separate computers. If the
ers fail independently. Computers are repaired in the evening
operating computer fails, one of the two spares can be imme-
and each day is an independent trial.
diately switched online. After the second computer fails, the
(a) What is the probability that all eight computers fail in a
last computer can be immediately switched online. Assume
day? 8
that the probability of a failure during any transaction is 10
(b) What is the mean number of days until a specific com- and that the transactions can be considered to be independent
puter fails? events.
(c) What is the mean number of days until all eight computers (a) What is the mean number of transactions before all com-
fail in the same day? puters have failed?
3-78. In Exercise 3-66, recall that 20 parts are checked each (b) What is the variance of the number of transactions before
hour and that X denotes the number of parts in the sample of all computers have failed?
20 that require rework.
3-85. Derive the expressions for the mean and variance of a
(a) If the percentage of parts that require rework remains at
geometric random variable with parameter p. (Formulas for
1%, what is the probability that hour 10 is the first sample
infinite series are required.)
at which X exceeds 1?
(b) If the rework percentage increases to 4%, what is the
probability that hour 10 is the first sample at which X
exceeds 1?