Page 99 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 77

3-6 BINOMIAL DISTRIBUTION 77

(c) Four identical electronic components are wired to a con- mial random variable with p 0.001. If 1000 bits are trans-
troller that can switch from a failed component to one of mitted, determine the following:
the remaining spares. Let X denote the number of compo- (a) P1X 12 (b) P1X 12
nents that have failed after a specified period of operation. (c) P1X 22 (d) mean and variance of X
(d) Let X denote the number of accidents that occur along the 3-64. The phone lines to an airline reservation system are
federal highways in Arizona during a one-month period. occupied 40% of the time. Assume that the events that the lines
(e) Let X denote the number of correct answers by a student are occupied on successive calls are independent. Assume that
taking a multiple choice exam in which a student can elim- 10 calls are placed to the airline.
inate some of the choices as being incorrect in some ques- (a) What is the probability that for exactly three calls the lines
tions and all of the incorrect choices in other questions. are occupied?
(f) Defects occur randomly over the surface of a semiconduc- (b) What is the probability that for at least one call the lines
tor chip. However, only 80% of defects can be found by are not occupied?
testing. A sample of 40 chips with one defect each is (c) What is the expected number of calls in which the lines
tested. Let X denote the number of chips in which the test are all occupied?
finds a defect.
(g) Reconsider the situation in part (f). Now, suppose the sam- 3-65. Batches that consist of 50 coil springs from a production
ple of 40 chips consists of chips with 1 and with 0 defects. process are checked for conformance to customer requirements.
(h) A filling operation attempts to fill detergent packages to The mean number of nonconforming coil springs in a batch is 5.
the advertised weight. Let X denote the number of deter- Assume that the number of nonconforming springs in a batch,
gent packages that are underfilled. denoted as X, is a binomial random variable.
(i) Errors in a digital communication channel occur in bursts (a) What are n and p?
that affect several consecutive bits. Let X denote the num- (b) What is P1X 22 ?
ber of bits in error in a transmission of 100,000 bits. (c) What is P1X 492 ?
(j) Let X denote the number of surface flaws in a large coil of 3-66. A statistical process control chart example. Samples
galvanized steel. of 20 parts from a metal punching process are selected every
3-56. The random variable X has a binomial distribution with hour. Typically, 1% of the parts require rework. Let X denote
n 10 and p 0.5. Sketch the probability mass function of X. the number of parts in the sample of 20 that require rework. A
(a) What value of X is most likely? process problem is suspected if X exceeds its mean by more
(b) What value(s) of X is(are) least likely? than three standard deviations.
3-57. The random variable X has a binomial distribution with (a) If the percentage of parts that require rework remains at
1%, what is the probability that X exceeds its mean by
n 10 and p 0.5. Determine the following probabilities: more than three standard deviations?
(a) P1X 52 (b) P1X 22
(b) If the rework percentage increases to 4%, what is the
(c) P1X 92 (d) P13 X 52 probability that X exceeds 1?
3-58. Sketch the probability mass function of a binomial (c) If the rework percentage increases to 4%, what is the
distribution with n 10 and p 0.01 and comment on the probability that X exceeds 1 in at least one of the next five
shape of the distribution. hours of samples?
(a) What value of X is most likely? Because not all airline passengers show up for their
(b) What value of X is least likely? 3-67.
reserved seat, an airline sells 125 tickets for a flight that holds
3-59. The random variable X has a binomial distribution with only 120 passengers. The probability that a passenger does not
n 10 and p 0.01. Determine the following probabilities. show up is 0.10, and the passengers behave independently.
(a) P1X 52 (b) P1X 22 (a) What is the probability that every passenger who shows
(c) P1X 92 (d) P13 X 52 up can take the flight?
3-60. Determine the cumulative distribution function of a (b) What is the probability that the flight departs with empty
binomial random variable with n 3 and p 1 2. seats?
3-61. Determine the cumulative distribution function of a 3-68. This exercise illustrates that poor quality can affect
binomial random variable with n 3 and p 1 4. schedules and costs. A manufacturing process has 100 cus-
3-62. An electronic product contains 40 integrated circuits. tomer orders to fill. Each order requires one component part
The probability that any integrated circuit is defective is 0.01, that is purchased from a supplier. However, typically, 2% of
and the integrated circuits are independent. The product oper- the components are identified as defective, and the compo-
ates only if there are no defective integrated circuits. What is nents can be assumed to be independent.
the probability that the product operates? (a) If the manufacturer stocks 100 components, what is the
3-63. Let X denote the number of bits received in error in a probability that the 100 orders can be filled without
digital communication channel, and assume that X is a bino- reordering components?