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applied statistics and probability for engineers
Applied Statistics And Probability For Engineers 3rd Edition Douglas C. Montgomery, George C. Runger - Solutions
Suppose that contamination particle size (in micrometers)can be modeled as for Determine the mean of X.
The thickness of a conductive coating in micrometers has a density function of 600x2 for 100 m x 120 m.(a) Determine the mean and variance of the coating thickness.(b) If the coating costs $0.50 per micrometer of thickness on each part, what is the average cost of the coating per part?
Determine the mean and variance of the weight of packages in Exercise 4.7..
The gap width is an important property of a magnetic recording head. In coded units, if the width is a continuous random variable over the range from 0 x 2 with f(x) 0.5x, determine the cumulative distribution function of the gap width.
Determine the cumulative distribution function for the distribution in Exercise 4-8.. Use the cumulative distribution function to determine the probability that a length exceeds 75 millimeters.
Determine the cumulative distribution function for the distribution in Exercise 4-6.. Use the cumulative distribution function to determine the probability that a component lasts more than 3000 hours before failure.
Determine the cumulative distribution function for the distribution in Exercise 4-4..
Determine the cumulative distribution function for the distribution in Exercise 4-3..
Determine the cumulative distribution function for the distribution in Exercise 4-1..
A manufacturer stocks components obtained from a supplier. Suppose that 2% of the components are defective and that the defective components occur independently.How many components must the manufacturer have in stock so that the probability that 100 orders can be completed without reordering
A large bakery can produce rolls in lots of either 0, 1000, 2000, or 3000 per day. The production cost per item is $0.10. The demand varies randomly according to the following distribution:demand for rolls 0 1000 2000 3000 probability of demand 0.3 0.2 0.3 0.2 Every roll for which there is a demand
Surface flaws in automobile exterior panels follow a Poisson distribution with a mean of 0.1 flaw per panel. If 100 panels are checked, what is the probability that fewer than five panels have any flaws?
A company performs inspection on shipments from suppliers in order to detect nonconforming products.The company’s policy is to use a sample size that is always 10% of the lot size. Comment on the effectiveness of this policy as a general rule for all sizes of lots.
A company performs inspection on shipments from suppliers in order to defect nonconforming products.Assume a lot contains 1000 items and 1% are nonconforming. What sample size is needed so that the probability of choosing at least one nonconforming item in the sample is at least 0.90? Assume the
Derive the formula for the mean and standard deviation of a discrete uniform random variable over the range of integers .
Show that the function f (x) in Example 3-5 satisfies the properties of a probability mass function by summing the infinite series.
Derive the mean and variance of a hypergeometric random variable (difficult exercise).
Flaws occur in the interior of plastic used for automobiles according to a Poisson distribution with a mean of 0.02 flaw per panel.(a) If 50 panels are inspected, what is the probability that there are no flaws?(b) What is the expected number of panels that need to be inspected before a flaw is
Messages arrive to a computer server according to a Poisson distribution with a mean rate of 10 per hour. Determine the length of an interval of time such that the probability that no messages arrive during this interval is 0.90.
It is suspected that some of the totes containing chemicals purchased from a supplier exceed the moisture content target. Samples from 30 totes are to be tested for moisture content. Assume that the totes are independent. Determine the proportion of totes from the supplier that must exceed the
From 500 customers, a major appliance manufacturer will randomly select a sample without replacement. The company estimates that 25% of the customers will provide useful data. If this estimate is correct, what is the probability mass function of the number of customers that will provide useful
An installation technician for a specialized communication system is dispatched to a city only when three or more orders have been placed. Suppose orders follow a Poisson distribution with a mean of 0.25 per week for a city with a population of 100,000 and suppose your city contains a population of
Assume the number of errors along a magnetic recording surface is a Poisson random variable with a mean of one error every bits. A sector of data consists of 4096 eight-bit bytes.(a) What is the probability of more than one error in a sector?(b) What is the mean number of sectors until an error is
Each main bearing cap in an engine contains four bolts. The bolts are selected at random, without replacement, from a parts bin that contains 30 bolts from one supplier and 70 bolts from another.(a) What is the probability that a main bearing cap contains all bolts from the same supplier?(b) What
Determine the probability mass function for the random variable with the following cumulative distribution function: 0.2 F(x) = 0.5 0.8 1 x
The random variable X has the following probability distribution:x 2 3 5 8 probability 0.2 0.4 0.3 0.1 Determine the following: (a) P(X 3) (b) P(X > 2.5) (c) P(2.7
Messages that arrive at a service center for an information systems manufacturer have been classified on the basis of the number of keywords (used to help route messages) and the type of message, either email or voice. Also, 70% of the messages arrive via email and the rest are voice.number of
Amanufacturer of a consumer electronics product expects 2% of units to fail during the warranty period. Asample of 500 independent units is tracked for warranty performance.(a) What is the probability that none fails during the warranty period?(b) What is the expected number of failures during the
Determine the constant c so that the following function is a probability mass function: for x 1, 2, 3, 4.
Determine the minimum number of assemblies that need to be checked so that the probability of at least one defective assembly exceeds 0.95.
Continuation of Exercise
In a manufacturing process that laminates several ceramic layers, 1% of the assemblies are defective. Assume that the assemblies are independent.(a) What is the mean number of assemblies that need to be checked to obtain five defective assemblies?(b) What is the standard deviation of the number of
Patient response to a generic drug to control pain is scored on a 5-point scale, where a 5 indicates complete relief.Historically the distribution of scores is 1 2 3 4 5 0.05 0.1 0.2 0.25 0.4 Two patients, assumed to be independent, are each scored.(a) What is the probability mass function of the
The probability that an individual recovers from an illness in a one-week time period without treatment is 0.1.Suppose that 20 independent individuals suffering from this illness are treated with a drug and 4 recover in a one-week time period. If the drug has no effect, what is the probability that
The number of errors in a textbook follow a Poisson distribution with a mean of 0.01 error per page. What is the probability that there are three or less errors in 100 pages?
A Web site is operated by four identical computer servers. Only one is used to operate the site; the others are spares that can be activated in case the active server fails. The probability that a request to the Web site generates a failure in the active server is 0.0001. Assume that each request
The number of messages sent to a computer bulletin board is a Poisson random variable with a mean of 5 messages per hour.(a) What is the probability that 5 messages are received in 1 hour?(b) What is the probability that 10 messages are received in 1.5 hours?(c) What is the probability that less
Continuation of Exercise 3-109.(a) What is the probability that you must call six times in order for two of your calls to be answered in less than 30 seconds?(b) What is the mean number of calls to obtain two answers in less than 30 seconds?
Continuation of Exercise 3-108.(a) What is the probability that you must call four times to obtain the first answer in less than 30 seconds?(b) What is the mean number of calls until you are answered in less than 30 seconds?
The probability that your call to a service line is answered in less than 30 seconds is 0.75. Assume that your calls are independent.(a) If you call 10 times, what is the probability that exactly 9 of your calls are answered within 30 seconds?(b) If you call 20 times, what is the probability that
Ashipment of chemicals arrives in 15 totes. Three of the totes are selected at random, without replacement, for an inspection of purity. If two of the totes do not conform to purity requirements, what is the probability that at least one of the nonconforming totes is selected in the sample?
The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.02 failure per hour.(a) What is the probability that the instrument does not fail in an 8-hour shift?(b) What is the probability of at least one failure in a
The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.05 flaw per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel.(a) What is the probability that there are no surface flaws
The number of failures for a cytogenics machine from contamination in biological samples is a Poisson random variable with a mean of 0.01 per 100 samples.(a) If the lab usually processes 500 samples per day, what is the expected number of failures per day?(b) What is the probability that the
The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile.(a) What is the probability that there are no cracks that require repair in 5 miles of highway?(b) What is the
When a computer disk manufacturer tests a disk, it writes to the disk and then tests it using a certifier. The certifier counts the number of missing pulses or errors. The number of errors on a test area on a disk has a Poisson distribution with 0.2.(a) What is the expected number of errors per
The number of flaws in bolts of cloth in textile manufacturing is assumed to be Poisson distributed with a mean of 0.1 flaw per square meter.(a) What is the probability that there are two flaws in 1 square meter of cloth?(b) What is the probability that there is one flaw in 10 square meters of
The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable.Assume that on the average there are 10 calls per hour.(a) What is the probability that there are exactly 5 calls in one hour?(b) What is the probability that there are 3 or less calls in one
Suppose that the number of customers that enter a bank in an hour is a Poisson random variable, and suppose that Determine the mean and variance of X.
Suppose X has a Poisson distribution with a mean of 0.4. Determine the following probabilities: (a) P(X = 0) (c) P(X = 4) (b) P(X 2) (d) P(X = 8)
Suppose X has a Poisson distribution with a mean of 4. Determine the following probabilities: (a) P(X = 0) (c) P(X = 4) (b) P(X 2) (d) P(X = 8)
What is the finite population correction in this exercise?
Use the binomial approximation to the hypergeometric distribution to approximate the probabilities in Exercise
For which exercise should the binomial approximation to the distribution of X be better?(b) For Exercise 3-86, calculate and assuming that X has a binomial distribution and compare these results to results derived from the hypergeometric distribution.(c) For Exercise 3-87, calculate and assuming
Continuation of Exercises 3-86 and 3-87.(a) Calculate the finite population corrections for Exercises 3-86 and
A state runs a lottery in which 6 numbers are randomly selected from 40, without replacement. A player chooses 6 numbers before the state’s sample is selected.(a) What is the probability that the 6 numbers chosen by a player match all 6 numbers in the state’s sample?(b) What is the probability
Magnetic tape is slit into half-inch widths that are wound into cartridges. A slitter assembly contains 48 blades.Five blades are selected at random and evaluated each day for sharpness. If any dull blade is found, the assembly is replaced with a newly sharpened set of blades.(a) If 10 of the
Printed circuit cards are placed in a functional test after being populated with semiconductor chips. Alot contains 140 cards, and 20 are selected without replacement for functional testing.(a) If 20 cards are defective, what is the probability that at least 1 defective card is in the sample?(b) If
A company employs 800 men under the age of 55.Suppose that 30% carry a marker on the male chromosome that indicates an increased risk for high blood pressure.(a) If 10 men in the company are tested for the marker in this chromosome, what is the probability that exactly 1 man has the marker?(b) If
Alot of 75 washers contains 5 in which the variability in thickness around the circumference of the washer is unacceptable.A sample of 10 washers is selected at random, without replacement.(a) What is the probability that none of the unacceptable washers is in the sample?(b) What is the probability
Determine the cumulative distribution function for X in Exercise 3-88.
Suppose X has a hypergeometric distribution with N 10, n 3, and K 4. Sketch the probability mass function of X.
Suppose X has a hypergeometric distribution with N 20, n 4, and K 4. Determine the following:(a) (b)(c) (d) Determine the mean and variance of X.
Suppose X has a hypergeometric distribution with N 100, n 4, and K 20. Determine the following:(a) (b)(c) (d) Determine the mean and variance of X.
Derive the expressions for the mean and variance of a geometric random variable with parameter p. (Formulas for infinite series are required.)
A fault-tolerant system that processes transactions for a financial services firm uses three separate computers. If the operating computer fails, one of the two spares can be immediately switched online. After the second computer fails, the last computer can be immediately switched online. Assume
An electronic scale in an automated filling operation stops the manufacturing line after three underweight packages are detected. Suppose that the probability of an underweight package is 0.001 and each fill is independent.(a) What is the mean number of fills before the line is stopped?(b) What is
The probability is 0.6 that a calibration of a transducer in an electronic instrument conforms to specifications for the measurement system. Assume the calibration attempts are independent. What is the probability that at most three calibration attempts are required to meet the specifications for
Suppose that X is a negative binomial random variable with p 0.2 and r 4. Determine the following:(a) (b)(c) (d)(e) The most likely value for X
Show that the probability density function of a negative binomial random variable equals the probability density function of a geometric random variable when r 1. Show that the formulas for the mean and variance of a negative binomial random variable equal the corresponding results for geometric
Consider a sequence of independent Bernoulli trials with p 0.2.(a) What is the expected number of trials to obtain the first success?(b) After the eighth success occurs, what is the expected number of trials to obtain the ninth success?
In Exercise 3-66, recall that 20 parts are checked each hour and that X denotes the number of parts in the sample of 20 that require rework.(a) If the percentage of parts that require rework remains at 1%, what is the probability that hour 10 is the first sample at which X exceeds 1?(b) If the
A trading company has eight computers that it uses to trade on the New York Stock Exchange (NYSE). The probability of a computer failing in a day is 0.005, and the computers fail independently. Computers are repaired in the evening and each day is an independent trial.(a) What is the probability
In Exercise 3-70, recall that a particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents an independent trial.(a) What is the probability that the first morning that the light is green is the fourth morning that you
Assume that each of your calls to a popular radio station has a probability of 0.02 of connecting, that is, of not obtaining a busy signal. Assume that your calls are independent.(a) What is the probability that your first call that connects is your tenth call?(b) What is the probability that it
In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene is 0.1.(a) What is the probability 4 or more people will have to be tested before 2 with the gene are detected?(b) How many people are
The probability of a successful optical alignment in the assembly of an optical data storage product is 0.8. Assume the trials are independent.(a) What is the probability that the first successful alignment requires exactly four trials?(b) What is the probability that the first successful alignment
Suppose the random variable X has a geometric distribution with a mean of 2.5. Determine the following probabilities: (a) P(X1) (b) P(X = 4) (c) P(X5) (d) P(X 3) (e) P(X3)
Suppose the random variable X has a geometric distribution with p 0.5. Determine the following probabilities: (a) P(X1) (c) P(X = 8) (e) P(X (b) P(X = 4) (d) P(X 2) 2)
Aparticularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents an independent trial.(a) Over five mornings, what is the probability that the light is green on exactly one day?(b) Over 20 mornings, what is the probability
A multiple choice test contains 25 questions, each with four answers. Assume a student just guesses on each question.(a) What is the probability that the student answers more than 20 questions correctly?(b) What is the probability the student answers less than 5 questions correctly?
This exercise illustrates that poor quality can affect schedules and costs. A manufacturing process has 100 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, 2% of the components are identified as defective, and the components can
Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently.(a) What is the probability that every passenger who shows
A statistical process control chart example. Samples of 20 parts from a metal punching process are selected every hour. Typically, 1% of the parts require rework. Let X denote the number of parts in the sample of 20 that require rework. A process problem is suspected if X exceeds its mean by more
Batches that consist of 50 coil springs from a production process are checked for conformance to customer requirements.The mean number of nonconforming coil springs in a batch is 5.Assume that the number of nonconforming springs in a batch, denoted as X, is a binomial random variable.(a) What are n
The phone lines to an airline reservation system are occupied 40% of the time. Assume that the events that the lines are occupied on successive calls are independent. Assume that 10 calls are placed to the airline.(a) What is the probability that for exactly three calls the lines are occupied?(b)
Let X denote the number of bits received in error in a digital communication channel, and assume that X is a bino- mial random variable with p 0.001. If 1000 bits are transmitted, determine the following:(a) (b)(c) (d) mean and variance of X
An electronic product contains 40 integrated circuits.The probability that any integrated circuit is defective is 0.01, and the integrated circuits are independent. The product operates only if there are no defective integrated circuits. What is the probability that the product operates?
Determine the cumulative distribution function of a binomial random variable with n 3 and p 14.
Determine the cumulative distribution function of a binomial random variable with n 3 and p 12.
The random variable X has a binomial distribution with n 10 and p 0.01. Determine the following probabilities.(a) (b)(c) (d)
Sketch the probability mass function of a binomial distribution with n 10 and p 0.01 and comment on the shape of the distribution.(a) What value of X is most likely?(b) What value of X is least likely?
The random variable X has a binomial distribution with n 10 and p 0.5. Determine the following probabilities:(a) (b)(c) (d)
The random variable X has a binomial distribution with n 10 and p 0.5. Sketch the probability mass function of X.(a) What value of X is most likely?(b) What value(s) of X is(are) least likely?
For each scenario described below, state whether or not the binomial distribution is a reasonable model for the random variable and why. State any assumptions you make.(a) A production process produces thousands of temperature transducers. Let X denote the number of nonconforming transducers in a
The probability of an operator entering alphanumeric data incorrectly into a field in a database is equally likely. The random variable X is the number of fields on a data entry form with an error. The data entry form has 28 fields. Is X a discrete uniform random variable? Why or why not.
Show that for a discrete uniform random variable X, if each of the values in the range of X is multiplied by the constantc, the effect is to multiply the mean of X by c and the variance of X by . That is, show that and .
Suppose that X has a discrete uniform distribution on the integers 0 through 9. Determine the mean, variance, and standard deviation of the random variable Y 5X and compare to the corresponding results for X.
The lengths of plate glass parts are measured to the nearest tenth of a millimeter. The lengths are uniformly distributed, with values at every tenth of a millimeter starting at 18 14 38 1 x 3 0 x 100 590.0 and continuing through 590.9. Determine the mean and variance of lengths.
Product codes of 2, 3, or 4 letters are equally likely.What is the mean and standard deviation of the number of letters in 100 codes?
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