Page 161 - Applied Statistics And Probability For Engineers

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4-12 LOGNORMAL DISTRIBUTION 139

4-145. A square inch of carpeting contains 50 carpet fibers. for the flight is 0.9 and the passengers are assumed to be inde-
The probability of a damaged fiber is 0.0001. Assume the pendent.
damaged fibers occur independently. (a) Approximate the probability that all the passengers that
(a) Approximate the probability of one or more damaged arrive can be seated.
fibers in 1 square yard of carpeting. (b) Approximate the probability that there are empty seats.
(b) Approximate the probability of four or more damaged (c) Approximate the number of reservations that the airline
fibers in 1 square yard of carpeting. should make so that the probability that everyone who ar-
4-146. An airline makes 200 reservations for a flight that rives can be seated is 0.95. [Hint: Successively try values
holds 185 passengers. The probability that a passenger arrives for the number of reservations.]

MIND-EXPANDING EXERCISES

4-147. The steps in this exercise lead to the probabil- amplifiers have a mean of 20,000 hours and the remain-
ity density function of an Erlang random variable X with ing amplifiers have a mean of 50,000 hours, what pro-
parameters and r, f 1x2 x r r 1 x 1r 12!, x 0, portion of the amplifiers fail before 60,000 hours?
e
r 1, 2, p . 4-151. Lack of Memory Property. Show that for
(a) Use the Poisson distribution to express P1X x2 . an exponential random variable X, P1X t 1
t 2 0
(b) Use the result from part (a) to determine the cumu- X t 1 2 P1X t 2 2
lative distribution function of X. 4-152. A process is said to be of six-sigma quality if
(c) Differentiate the cumulative distribution function in the process mean is at least six standard deviations from
part (b) and simplify to obtain the probability den- the nearest specification. Assume a normally distributed
sity function of X. measurement.
4-148. A bearing assembly contains 10 bearings. The (a) If a process mean is centered between the upper and
bearing diameters are assumed to be independent and lower specifications at a distance of six standard de-
normally distributed with a mean of 1.5 millimeters and viations from each, what is the probability that a
a standard deviation of 0.025 millimeter. What is the product does not meet specifications? Using the
probability that the maximum diameter bearing in the result that 0.000001 equals one part per million,
assembly exceeds 1.6 millimeters? express the answer in parts per million.
4-149. Let the random variable X denote a measure- (b) Because it is difficult to maintain a process mean
ment from a manufactured product. Suppose the target centered between the specifications, the probability
value for the measurement is m. For example, X could of a product not meeting specifications is often cal-
denote a dimensional length, and the target might be 10 culated after assuming the process shifts. If the
millimeters. The quality loss of the process producing process mean positioned as in part (a) shifts upward
the product is defined to be the expected value of by 1.5 standard deviations, what is the probability
$k1X m2 2 , where k is a constant that relates a devia- that a product does not meet specifications? Express
tion from target to a loss measured in dollars. the answer in parts per million.
(a) Suppose X is a continuous random variable with (c) Rework part (a). Assume that the process mean is
E1X2 m and V1X2 2 . What is the quality loss at a distance of three standard deviations.
of the process? (d) Rework part (b). Assume that the process mean is at
(b) Suppose X is a continuous random variable with a distance of three standard deviations and then
E1X2 and V1X2 2 . What is the quality loss shifts upward by 1.5 standard deviations.
of the process? (e) Compare the results in parts (b) and (d) and comment.
4-150. The lifetime of an electronic amplifier is mod-
eled as an exponential random variable. If 10% of the