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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
The weight of a small candy is normally distributed with a mean of 0.1 ounce and a standard deviation of 0.01 ounce. Suppose that 16 candies are placed in a package and that the weights are independent.(a) What are the mean and variance of package net weight?(b) What is the probability that the net
The length and width of panels used for interior doors(in inches) are denoted as X and Y, respectively. Suppose that X and Y are independent, continuous uniform random variables for 17.75 x 18.25 and 4.75 y 5.25, respectively.(a) By integrating the joint probability density function over
The weight of adobe bricks for construction is normally distributed with a mean of 3 pounds and a standard deviation of 0.25 pound. Assume that the weights of the bricks are independent and that a random sample of 25 bricks is chosen.(a) What is the probability that the mean weight of the sample is
Contamination problems in semiconductor manufacturing can result in a functional defect, a minor defect, or no defect in the final product. Suppose that 20, 50, and 30% of the contamination problems result in functional, minor, and no defects, respectively. Assume that the effects of 10
The lifetimes of six major components in a copier are independent exponential random variables with means of 8000, 10,000, 10,000, 20,000, 20,000, and 25,000 hours, respectively.(a) What is the probability that the lifetimes of all the components exceed 5000 hours?(b) What is the probability that
Suppose that X and Y are independent, continuous uniform random variables for 0 x 1 and 0 y 1. Use the joint probability density function to determine the probability that
Continuation of Exercise 5-102(a) What is the conditional distribution of the number of calls requiring five rings or more given that eight calls are answered in two rings or less?(b) What is the conditional expected number of calls requiring five rings or more given that eight calls are answered
To evaluate the technical support from a computer manufacturer, the number of rings before a call is answered by a service representative is tracked. Historically, 70% of the calls are answered in two rings or less, 25% are answered in three or four rings, and the remaining calls require five rings
Continuation of Exercise 5-100(a) What is the conditional distribution of the number of bolts rated low given that 16 bolts are rated high?(b) What is the conditional expected number of bolts rated low given that 16 bolts are rated high?(c) Are the numbers of bolts rated high and low independent
The backoff torque required to remove bolts in a steel plate is rated as high, moderate, or low. Historically, the probability of a high, moderate, or low rating is 0.6, 0.3, or 0.1, respectively. Suppose that 20 bolts are evaluated and that the torque ratings are independent.(a) What is the
The percentage of people given an antirheumatoid medication who suffer severe, moderate, or minor side effects are 10, 20, and 70%, respectively. Assume that people react independently and that 20 people are given the medication.Determine the following:(a) The probability that 2, 4, and 14 people
Show that the following function satisfies the properties of a joint probability mass function: X y f(x, y) 0 0 1/4 0 1 1/8 1 0 1/8 1 1 1/4 2 2 1/4
Assume that the weights of individuals are independent and normally distributed with a mean of 160 pounds and a standard deviation of 30 pounds. Suppose that 25 people squeeze into an elevator that is designed to hold 4300 pounds.(a) What is the probability that the load (total weight) exceeds the
The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Assume that the thickness is normally distributed and that the thicknesses of different wafers are independent.(a) Determine the probability that the average thickness of
Soft-drink cans are filled by an automated filling machine and the standard deviation is 0.5 fluid ounce. Assume that the fill volumes of the cans are independent, normal random variables.(a) What is the standard deviation of the average fill volume of 100 cans?(b) If the mean fill volume is 12.1
The length of A is normally distributed with a mean of 10 millimeters and a standard deviation of 0.1 millimeter. The thickness of parts B and C is normally distributed with a mean of 2 millimeters and a standard deviation of 0.05 millimeter. Assume all dimensions are independent.(a) Determine the
A U-shaped component is to be formed from the three parts A, B, and C. The picture is shown in Fig.
The width of a casing for a door is normally distributed with a mean of 24 inches and a standard deviation of 18 inch. The width of a door is normally distributed with a mean of 23 and 78 inches and a standard deviation of 116 inch. Assume independence.(a) Determine the mean and standard
In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate.The thickness of these layers is critical if specifications regarding the final color and intensity of light of the lamp are to be met. Let X and Y denote the thickness of two
A plastic casing for a magnetic disk is composed of two halves. The thickness of each half is normally distributed with a mean of 2 millimeters and a standard deviation of 0.1 millimeter and the halves are independent.(a) Determine the mean and standard deviation of the total thickness of the two
Suppose that the random variable X represents the length of a punched part in centimeters. Let Y be the length of the part in millimeters. If E(X) 5 and V(X) 0.25, what are the mean and variance of Y?
If X and Y are independent, normal random variables with E(X) 0, V(X) 4, E(Y ) 10, and V(Y ) 9.Determine the following: (b) V(2X+ 3Y) (a) E(2X+ 3Y) (c) P(2X+3Y
If X and Y have a bivariate normal distribution with joint probability density fXY (x, y; X, Y, X, Y, ), show that the correlation between X and Y is . [Hint: Complete the square in the exponent].
If X and Y have a bivariate normal distribution with joint probability density fXY (x, y; X, Y, X, Y, ), show that the marginal probability distribution of X is normal with mean X and standard deviation X. [Hint: Complete the square in the exponent and use the fact that the integral of a
Show that the probability density function fXY (x, y;X, Y, X, Y, ) of a bivariate normal distribution integrates to one. [Hint: Complete the square in the exponent and use the fact that the integral of a normal probability density function for a single variable is 1.]
If X and Y have a bivariate normal distribution with 0, show that X and Y are independent.
Suppose that X and Y have a bivariate normal distribution with joint probability density function fXY (x, y; X, Y,X, Y, ).(a) Show that the conditional distribution of Y, given that X x is normal.(b) Determine .(c) Determine .
In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final color and intensity of light of the lamp are to be met. Let X and Y denote the thickness of two
Let X and Y represent two dimensions of an injection molded part. Suppose X and Y have a bivariate normal distribution with X 0.04, Y 0.08, X 3.00,Y 7.70, and Y 0. Determine P(2.95 X 3.05, 7.60 Y 7.80).
Let X and Y represent concentration and viscosity of a chemical product. Suppose X and Y have a bivariate normal distribution with X 4, Y 1, X 2, and Y 1. Draw a rough contour plot of the joint probability density function for each of the following values for :(a) 0 (b) 0.8(c)
Suppose X and Y are independent continuous random variables. Show that XY 0.
The joint probability distribution is x 1 0 1 y 1 1 0 fXY (x, y) 1 4 1 4 1 4 1 4 Show that the correlation between X and Y is zero, but X and Y are not independent.
Suppose that the correlation between X and Y is . For constantsa, b,c, andd, what is the correlation between the random variables U aX b and V cY d?
Determine the covariance and correlation for the joint probability density function over the range 0 x and 0 y.
Determine the covariance and correlation for the joint probability density function fXY (x, y) 6 106e0.001x0.002y over the range 0 x and x y from Example 5-15.
Determine the value for c and the covariance and correlation for the joint probability density function fXY (x, y) c over the range 0 x 5, 0 y, and x 1 y x 1.
Determine the value for c and the covariance and correlation for the joint probability density function fXY (x, y) cxy over the range 0 x 3 and 0 y x.
Determine the covariance and correlation for X1 and X2 in the joint distribution of the multinomial random variables X1, X2 and X3 in with p1 p2 p3 13 and n 3. What can you conclude about the sign of the correlation between two random variables in a multinomial distribution?
Determine the covariance and correlation for the joint probability distribution shown in Fig. 5-4(a) and described in Example 5-8.
Determine the value for c and the covariance and correlation for the joint probability mass function fXY (x, y) c(x y) for x 1, 2, 3 and y 1, 2, 3.
Determine the covariance and correlation for the following joint probability distribution:x 1 0.5 0.5 1 y 2 1 1 2 fXY (x, y) 1 8 1 4 1 2 1 8
Determine the covariance and correlation for the following joint probability distribution:x 1 1 2 4 y 3 4 5 6 fXY (x, y) 18 14 12 18
A manufacturer of electroluminescent lamps knows that the amount of luminescent ink deposited on one of its products is normally distributed with a mean of 1.2 grams and a standard deviation of 0.03 grams. Any lamp with less than 1.14 grams of luminescent ink will fail to meet customer’s
The weights of adobe bricks used for construction are normally distributed with a mean of 3 pounds and a standard deviation of 0.25 pound. Assume that the weights of the bricks are independent and that a random sample of 20 bricks is selected.(a) What is the probability that all the bricks in the
The yield in pounds from a day’s production is normally distributed with a mean of 1500 pounds and standard deviation of 100 pounds. Assume that the yields on different days are independent random variables.(a) What is the probability that the production yield exceeds 1400 pounds on each of five
Continuation of Exercise 5-6.. Determine the following:(a) Marginal distribution of X(b) Joint distribution of X and Y(c) Conditional probability distribution of X given that Y 0.5 and Z 0.5 (d) Conditional probability distribution of X given that Y 0.5
The conditional probability distribution of Y given X xis for y 0 and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10. (a) Graph fyx(y) = xey for y> 0 for several values of x. Determine (b) P(Y
A popular clothing manufacturer receives Internet orders via two different routing systems. The time between orders for each routing system in a typical day is known to be exponentially distributed with a mean of 3.2 minutes. Both systems operate independently.(a) What is the probability that no
The time between surface finish problems in a galvanizing process is exponentially distributed with a mean of 40 hours. A single plant operates three galvanizing lines that are assumed to operate independently.(a) What is the probability that none of the lines experiences a surface finish problem
Two methods of measuring surface smoothness are used to evaluate a paper product. The measurements are recorded as deviations from the nominal surface smoothness in coded units. The joint probability distribution of the two measurements is a uniform distribution over the region 0 x 4, 0 y,
Determine the following: (a) P(X
Continuation of Exercise
Determine the value of c such that the function f (x, y) cxy for 0 x 3 and 0 y 3 satisfies the properties of a joint probability density function.
A marketing company performed a risk analysis for a manufacturer of synthetic fibers and concluded that new competitors present no risk 13% of the time (due mostly to the diversity of fibers manufactured), moderate risk 72% of the time(some overlapping of products), and very high risk (competitor
Continuation of Exercise 5-30.. Let X and Y denote the number of bits with high and moderate distortion out of the three transmitted, respectively. Determine the following:(a) The probability distribution, mean and variance of X.(b) The conditional probability distribution, conditional mean and
In the transmission of digital information, the probability that a bit has high, moderate, or low distortion is 0.01, 0.04, and 0.95, respectively. Suppose that three bits are transmitted and that the amount of distortion of each bit is assumed to be independent.(a) What is the probability that two
Continuation of Exercise 5-27.. Determine the following:(a) The conditional probability that two ovens have major defects given that two ovens have minor defects(b) The conditional probability that three ovens have major defects given that two ovens have minor defects(c) The conditional probability
Determine the following:(a) The joint probability mass function of the number of ovens with a major defect and the number with a minor defect.(b) The expected number of ovens with a major defect.(c) The expected number of ovens with a minor defect.
Continuation of Exercise
Four electronic ovens that were dropped during shipment are inspected and classified as containing either a major, a minor, or no defect. In the past, 60% of dropped ovens had a major defect, 30% had a minor defect, and 10% had no defect. Assume that the defects on the four ovens occur
Continuation of Exercise 5-23.. Determine the conditional probability distribution of X given that Y 2.
An order of 15 printers contains four with a graphicsenhancement feature, five with extra memory, and six with both features. Four printers are selected at random, without replacement, from this set. Let the random variables X, Y, and Z denote the number of printers in the sample with graphics
Based on the number of voids, a ferrite slab is classified as either high, medium, or low. Historically, 5% of the slabs are classified as high, 85% as medium, and 10% as low.A sample of 20 slabs is selected for testing. Let X, Y, and Z denote the number of slabs that are independently classified
Continuation of Exercise 5-17.. Determine the conditional probability distribution of X given that Y 1 and Z 2.
Suppose the random variables X, Y, and Z have the following joint probability distribution x y z f(x, y, z)1 1 1 0.05 1 1 2 0.10 1 2 1 0.15 1 2 2 0.20 2 1 1 0.20 2 1 2 0.15 2 2 1 0.10 2 2 2 0.05
A manufacturing company employs two inspecting devices to sample a fraction of their output for quality control purposes. The first inspection monitor is able to accurately detect 99.3% of the defective items it receives, whereas the second is able to do so in 99.7% of the cases. Assume that four
A small-business Web site contains 100 pages and 60%, 30%, and 10% of the pages contain low, moderate, and high graphic content, respectively. A sample of four pages is selected without replacement, and X and Y denote the number of pages with moderate and high graphics output in the sample.
In the transmission of digital information, the probability that a bit has high, moderate, and low distortion is 0.01, 0.10, and 0.95, respectively. Suppose that three bits are transmitted and that the amount of distortion of each bit is assumed to be independent.Let X and Y denote the number of
Four electronic printers are selected from a large lot of damaged printers. Each printer is inspected and classified as containing either a major or a minor defect. Let the random variables X and Y denote the number of printers with major and minor defects, respectively. Determine the range of the
Show that the following function satisfies the properties of a joint probability mass function.x y fXY (x, y)1 1 14 1.5 2 18 1.5 3 14 2.5 4 14 3 5 18
A process is said to be of six-sigma quality if the process mean is at least six standard deviations from the nearest specification. Assume a normally distributed measurement.(a) If a process mean is centered between the upper and lower specifications at a distance of six standard deviations from
Lack of Memory Property. Show that for an exponential random variable X,
The lifetime of an electronic amplifier is modeled as an exponential random variable. If 10% of the amplifiers have a mean of 20,000 hours and the remaining amplifiers have a mean of 50,000 hours, what proportion of the amplifiers fail before 60,000 hours?
Let the random variable X denote a measurement from a manufactured product. Suppose the target value for the measurement is m. For example, X could denote a dimensional length, and the target might be 10 millimeters. The quality loss of the process producing the product is defined to be the
A bearing assembly contains 10 bearings. The bearing diameters are assumed to be independent and normally distributed with a mean of 1.5 millimeters and a standard deviation of 0.025 millimeter. What is the probability that the maximum diameter bearing in the assembly exceeds 1.6 millimeters?
The steps in this exercise lead to the probability density function of an Erlang random variable X with parameters and(a) Use the Poisson distribution to express .(b) Use the result from part (a) to determine the cumulative distribution function of X.(c) Differentiate the cumulative distribution
An airline makes 200 reservations for a flight that holds 185 passengers. The probability that a passenger arrives for the flight is 0.9 and the passengers are assumed to be independent.(a) Approximate the probability that all the passengers that arrive can be seated.(b) Approximate the probability
A square inch of carpeting contains 50 carpet fibers.The probability of a damaged fiber is 0.0001. Assume the damaged fibers occur independently.(a) Approximate the probability of one or more damaged fibers in 1 square yard of carpeting.(b) Approximate the probability of four or more damaged fibers
Rework parts (a)and (b). Assume that the lifetime is a lognormal random variable with the same mean and standard deviation.
Continuation of Exercise
Continuation of Exercise 140. Rework parts (a) and(b). Assume that the lifetime is an exponential random variable with the same mean.
A product contains three lasers, and the product fails if any of the lasers fails.Assume the lasers fail independently. What should the mean life equal in order for 99% of the products to exceed 10,000 hours before failure?
Continuation of Exercise
What should the mean life equal in order for 99% of the lasers to exceed 10,000 hours before failure?
Continuation of Exercise
The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours.(a) What is the probability that a laser fails before 5,800 hours?(b) What is the life in hours that 90% of the lasers exceed?
Assume that the standard deviation of the size of a dot is 0.0004 inch. If the probability that a dot meets specifications is to be 0.9973, what specifications are needed? Assume that the specifications are to be chosen symmetrically around the mean of 0.002.
Continuation of Exercise
The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch.Suppose that the specifications require the dot diameter to be between 0.0014 and 0.0026 inch. If the probability that a dot meets specifications is to be 0.9973, what standard deviation is
The thickness of a laminated covering for a wood surface is normally distributed with a mean of 5 millimeters and a standard deviation of 0.2 millimeter.(a) What is the probability that a covering thickness is greater than 5.5 millimeters?(b) If the specifications require the thickness to be
With an automated irrigation system, a plant grows to a height of 3.5 centimeters two weeks after germination.(a) What is the probability of obtaining a plant of this height or greater from the distribution of heights in Exercise 4-135.(b) Do you think the automated irrigation system increases the
Continuation of Exercise
Without an automated irrigation system, the height of plants two weeks after germination is normally distributed with a mean of 2.5 centimeters and a standard deviation of 0.5 centimeters.(a) What is the probability that a plant’s height is greater than 2.25 centimeters?(b) What is the
Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per squared centimeter of surface dust is 100. A sample of 800 square centimeters of dust is
Suppose that X has a lognormal distribution and that the mean and variance of X are 50 and 4000, respectively.Determine the following:(a) The parameters and of the lognormal distribution(b) The probability that X is less than 150
Suppose that X has a lognormal distribution with parameters and . Determine the following:(a)(b) The value for x such that(c) The mean and variance of X
Assume that your corporation has owned 10 CPUs for three years, and assume that the CPUs fail independently. What is the probability that at least one fails within the next three years?
Continuation of Exercise
The CPU of a personal computer has a lifetime that is exponentially distributed with a mean lifetime of six years.You have owned this CPU for three years. What is the probability that the CPU fails in the next three years?
Continuation of Exercise 4-127.(a) What is the probability that the time until the third call is greater than 30 minutes?(b) What is the mean time until the fifth call?
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