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Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions

Continuation of Exercise 140 Rework parts (a) and (b). Assume that the lifetime is an exponential random variable with the same mean.
Continuation of Exercise 4-140. Rework parts (a) And (b). Assume that the lifetime is a lognormal random variable with the same mean and standard deviation.
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
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
The steps in this exercise lead to the probability density function of an Erlang random variable X with parameters λ and r, f(x) = λr xr–1 e–λ /(r – 1) !, x > 0, r = 1, 2, . .. (a) Use the Poisson distribution to express P(X > x). (b) Use the result from part (a) to determine
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?
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
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?
Lack of Memory Property Show that for an exponential random variable X, P(X t1) = P(X < t2)
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
Show that the following function satisfies the properties of a joint probability mass function.
Continuation of Exercise 5-1 Determine the following probabilities:(a) P(X < 2.5, Y < 3)(b) P(X < 2.5)(c) P(Y < 3)(d) P(X > 1.8, Y > 4.7)
Continuation of Exercise 5-1 Determine and E (X) and E(Y).
Continuation of Exercise 5-1 Determine. (a) The marginal probability distribution of the random variable X. (b) The conditional probability distribution of Y given that λ = 1.5. (c) The conditional probability distribution of X given that Y = 2. (d) E(Y| X = 1.5) (e) Are X and Y
Determine the value of c that makes the function f(x, y) = c (x + y) a joint probability mass function over the nine points with x = 1, 2, 3 and y = 1, 2, 3.
Continuation of Exercise 5-5 determine the following probabilities(a) P(X = 1, Y < 4)(b) P(X = 1)(c) P(Y = 2)(d) P(X < 2, Y < 2)
Continuation of Exercise 5-5 Determine E(X), E(Y), V(X), and V (Y).
Continuation of Exercise 5-5 Determine(a) The marginal probability distribution of the random variable X. (b) The conditional probability distribution of Y given that X = 1.(c) The conditional probability distribution of X given that Y = 2.(d) E(Y| X =1)(e) Are X and Y independent?
Show that the following function satisfies the properties of a joint probability mass function.
Continuation of Exercise 5-9. Determine the following probabilities:(a) P(X < 0.5 Y < 1.5)(b) P(Y < 1.5)(c) P(X < 0.5) (d) P(X > 0.25 Y < 4.5)
Continuation of Exercise 5-9. Determine E(X) and E(Y ).
Continuation of Exercise 5-9. Determine(a) The marginal probability distribution of the random variable X.(b) The conditional probability distribution of Y given that X = 1.(c) The conditional probability distribution of X given that Y = 1.(d) E(X |y = 1)(e) Are X and Y independent?
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
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
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.
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
Suppose the random variables X, Y, and Z have the following joint probability distributionDetermine the following:(a) P(X = 2)(b) P(X = 1, Y = 2)(c) P(Z (d) P(X = 1 or Z = 2)(e) E(X)
Continuation of Exercise 5-17. Determine the following:(a) P(X = 1|Y = 1)(b) P(X = 1, Y = 1|Z = 2)(c) P(X = 1|Y = 1, Z = 2)
Continuation of Exercise 5-17. Determine the conditional probability distribution of X given that Y = 1 and Z = 2.
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 as
Continuation of Exercise 5-20. Determine the following:(a) P(X = 1, Y = 17, Z = 3)(b) P(X < 1, Y = 17, Z = 3)(c) P(X < 1(d) E(X)
Continuation of Exercise 5-20. Determine the following:(a) P(X = 2, Z = 3|Y = 17)(b) P(X = 2|Y = 17)(c) E(X|Y = 17)
An order of 15 printers contains four with a graphics enhancement 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
Continuation of Exercise 5-23. Determine the conditional probability distribution of X given that Y = 2.
Continuation of Exercise 5-23. Determine the following:(a) P(X = 1, Y = 2, Z =1)(b) P(X = 1, Y = 1)(c) E(X) and V(X)
Continuation of Exercise 5-23. Determine the following:(a) P(X = 1, Y = 2|Z =1)(b) P(X = 2/Y = 2)(c) The conditional probability distribution of X given that Y = 0 and Z = 3.
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 5-27. 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 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
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 5-30. 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
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 5-32. Determine the following:(a) P(Z = 2|Y = 1, X = 10) (b) P(Z < 1|X = 10) (c) P(Z < 1, Z < = 1|X = 10) (d) E(Z|X = 10)
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.
Continuation of Exercise 5-34. Determine the following:(a) P(X < 2.Y < 3)(b) P(X < 2.5)(c) P(1 < Y< 2.5) (d) P(X > 1.8, 1 < Y < 2.5)(e) E(X) (f) P(X < 0, Y < 4)
Continuation of Exercise 5-34. Determine the following:(a) Marginal probability distribution of the random variable X(b) Conditional probability distribution of Y given that X = 1.5(c) E(Y|X) = 1/5)(d) P(Y < 2|X =1/5)(e) Conditional probability distribution of X given that Y = 2
Determine the value of c that makes the function f(x, y) = c(x = y) a joint probability density function over the range 0 < x < 3 and x < y < x + 2.
Continuation of Exercise 5-37. Determine the following:(a) P(X < 1, Y < 2)(b) P(1 < X < 2)(c) P(Y > 1)(d) P(X < 2, Y < 2)(e) E(X)
Continuation of Exercise 5-37. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given that X = 1(c) E(Y|X = 1)(d) P(Y > 2|X = 1)(e) Conditional probability distribution of X given that Y = 2
Determine the value of c that makes the function f(x, y) =cxy a joint probability density function over the range 0
Continuation of Exercise 5-40. Determine the following:(a) P(X < 1, Y< 2)(b) P(1 < X < 2)(c) P(Y > 1)(d) P(X < 2, Y< 2)(e) E(X) (f) E(Y)
Continuation of Exercise 5-40. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given X = 1(c) E(Y|X = 1)(d) P(Y > 2|X = 1)(e) Conditional probability distribution of X given Y = 2
Determine the value of c that makes the function f(x, y) = ce-2x – 3y a joint probability density function over the range 0 < x and 0 < y < x.
Continuation of Exercise 5-43. Determine the following:(a) P(X < 1, Y < 2)(b) P(1 < X, < 2)(c) P(Y > 3)(d) P(X
Continuation of Exercise 5-43. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given X = 1(c) E(Y|X = 1)(d) Conditional probability distribution of X given Y= 2
Determine the value of c that makes the function f(x, y) = ce-2x–3y a joint probability density function over the range 0 < x and x < y.
Continuation of Exercise 5-46. Determine the following:(a) P(X < 1, Y < 2)(b) P(1 < X, < 2)(c) P(Y > 2)(d) P(X < 2, Y < 2)(e) E(X)(f) E(Y)
Continuation of Exercise 5-46. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given X = 1(c) E(Y|X = 1)(d) P(Y
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,
Continuation of Exercise 5-49. Determine the following:(a) P(X
Continuation of Exercise 5-49. Determine the following:(a) Marginal probability distribution of X(b) Conditional probability distribution of Y given X = 1(c) X(Y|X = 1)(d) P(Y < 0.5|X = 1)
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
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 conditional probability distribution of Y given X = x is FY|X(y) = XE–xy for y > 0 and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10.(a) Graph is FY|X(y) = XE–xy for y > 0 for several values of x. Determine(b) P(Y < 2|X = 2)(c) E(Y|X = 2)(d)
Suppose the random variables X, Y, and Z have the joint probability density function f(x, y, z) = 8xyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Determine the following:(a) P(X < 0.5)(b) P(X < 0.5, Y < 0.5)(c) P(Z < 2)(d) P(X < 0.5, Z < 2)(e) E(X)
Continuation of Exercise 5-55. Determine the following:(a) P(X < 0.5|Y = 0.5)(b) P(X < 0.5, Y < 0.5|Z = 0.8)
Continuation of Exercise 5-55. Determine the following: (a) Conditional probability distribution of X given that Y = 0.5 and Z 0.8(b) P(X < 0.5|Y = 0.5, Z = 0.8)
Suppose the random variables X, Y, and Z have the joint probability density function fXYZ (x, y, z) = c over the cylinder x2 + y2 < 4 and 0 < z < 4. Determine the following.(a) The constant c so that fXYZ (x, y, z) is a probability density function(b) P(X2 + Y2 < 2)(c) P(Z < 2)(d) E(X)
Continuation of Exercise 5-58. Determine the following:(a) P(X < 1|Y = 1)(b) P(X2 + Y2 < 1|Z = 1)
Continuation of Exercise 5-58. Determine the conditional probability distribution of Z given that X = 1 and Y = 1.
Determine the value of c that makes fXYZ(x, y, z) = c a joint probability density function over the region x > 0, y > 0, z > 0, and x + y + z < 1.
Continuation of Exercise 5-61. Determine the following:(a) P(X < 0.5, Y < 0.5, z < 0.5)(b) P(X < 0.5, Y < 0.5(c) P(X < 0.5)(d) E(X)
Continuation of Exercise 5-61. 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 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
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
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
Determine the covariance and correlation for the following joint probability distribution:x 1 1 2 4y 3 4 5 6fXY (x, y) 1/8 1/4 1/2 1/8
Determine the covariance and correlation for the following joint probability distribution:x –1 –0.5 0.5 1y –2 –1 1 2 fXY (x, y) 1/8 ¼ ½ 1/8
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 joint probability distribution shown in Fig. 5-4(a) and described in Example 5-8.
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 = 1/3 and n = 3. What can you conclude about the sign of the correlation between two random variables in a multinomial distribution?
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 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 covariance and correlation for the joint probability density function fXY (x, y) = 6 X 10-6e –0.001x-0.002y over the range 0 < x and x < y from Example 5-15.
Determine the covariance and correlation for the joint probability density function fXY (x, y) = e–x–y over the range 0 < x and 0 < y.
Suppose that the correlation between X and Y is =. For constants a, b, c, and d, what is the correlation between the random variables U aX + b and V = cY + d?
The joint probability distribution is x - 1 0 0 1y 0 -1 1 0fXY (x, y) ¼ ¼ ¼ ¼ Show that the correlation between X and Y is zero, but X and Y are not independent.
Suppose X and Y are independent continuous random variables. Show that σXY = 0.
Let X and Y represent concentration and viscosity of a chemical product. Suppose X and Y have a bivariate normal distribution with σX = 4, σY = 1, μX = 2, and μY = 1. Draw a rough contour plot of the joint probability density function for each of the following values for p: (a)
Let X and Y represent two dimensions of an injection molded part. Suppose X and Y have a bivariate normal distribution with σX = 0.04, σY = 0.08, μX = 3.00, μY = 7.70, and pY = 0. Determine P(2.95 < X < 3.05, 7.60 < Y < 7.80).
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
Suppose that X and Y have a bivariate normal distribution with joint probability density function fXY (x, y; σX, σY, μX, μY, p). (a) Show that the conditional distribution of Y, given that X = x is normal. (b) Determine E(Y|X = x) (c) Determine V(Y|X = X)
If X and Y have a bivariate normal distribution with p = 0, show that X and Y are independent.
Show that the probability density function fXY (x, y; σX, σY, μX, μY, p) 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 joint probability density fXY (x, y; σX, σY, μX, μY, p), show that the marginal probability distribution of X is normal with mean μX and standard deviation σX. [Hint: Complete the square in the exponent and use the
If X and Y have a bivariate normal distribution with joint probability density fXY (x, y; σX, σY, μX, μY, p), show that the correlation between X and Y is p. [Hint: Complete the square in the exponent].
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:(a) E(2X + 3Y) (b) V(2X + 3Y)(c) P(2X + 3Y < 30)(d) P(2X + 3Y < 40)
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?
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
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

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