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Miller And Freunds Probability And Statistics For Engineers 9th Edition Richard A. Johnson - Solutions

5.120 A change is made to one product page on the retail companies’ web site. To determine if the change does improve the efficiency of that product page, data must be collected on the proportion of visitors to the new page that ultimately purchase the product. It is known that 3.2% of visitors,
5.119 Referring to the normal scores in Exercise 5.101, construct a normal scores plot of the current flow data in Exercise 2.68.
5.118 The probability density shown in Figure 5.21 is the exponential distribution f (x) = 0.55 e−0.55x 0 < x 0 elsewhere Find the probability that (a) the time to observe a particle is more than 200 microseconds; (b) the time to observe a particle is less than 10 microseconds.
5.116 Referring to Exercise 5.28, find the quartiles of the normal distribution with μ = 102 and σ = 27. 5.117 The probability density shown in Figure 5.19 is the log-normal distribution with α = 8.85 and β = 1.03. Find the probability that (a) the interrequest time is more than 200
5.115 Verify that (a) z0.10 = 1.28; (b) z0.001 = 3.09.
5.114 The burning time of an experimental rocket is a random variable having the normal distribution with μ = 4.76 seconds and σ = 0.04 second. What is the probability that this kind of rocket will burn (a) less than 4.66 seconds; (b) more than 4.80 seconds; (c) anywhere from 4.70 to 4.82 seconds?
5.113 Referring to Exercise 5.112, suppose the rotation speed of the coil can be increased and standard deviation decreased. Determine the new value for the standard deviation that would restrict the probability of an error greater than 0.085 V to be less than 0.02.
5.112 A coil is rotated in a magnetic field to generate current. The voltage generated can be modeled by a normal distribution having mean μ and standard deviation 0.5 V where μ is the true voltage. Find the probability that voltage generated will differ from the true voltage by (a) less than
5.111 In certain experiments, the error made in determining the density of a silicon compound is a random variable having the probability density f (x) = 25 for − 0.02 < x < 0.02 0 elsewhere Find the probabilities that such an error will be (a) between −0.03 and 0.04; (b) between −0.005 and
5.110 With reference to the preceding exercise, find the corresponding distribution function and use it to determine the probabilities that a random variable having this distribution function will take on a value (a) less than 0.3; (b) between 0.4 and 0.6.
5.109 If the probability density of a random variable is given by f (x) = k ( 1 − x2 ) for 0 < x < 1 0 elsewhere find the value of k and the probabilities that a random variable having this probability density will take on a value (a) between 0.1 and 0.2; (b) greater than 0.5. (c) Find μ and
5.108 The statistical package MINITAB has a normal random number generator. To simulate 5 values from a normal distribution having mean 7 and standard deviation 4, and place them in C1, use the commands Dialog Box: Calc > Random Sample > Normal Type 5 after Generate, C1 in Column, 7 in Mean and 4
5.107 The statistical package MINITAB has a random number generator. To simulate 5 values from an exponential distribution having mean β = 0.05, choose Dialog Box: Calc > Random Data > Exponential Type 5 after Generate, C1 in Column and 0.05 in Mean. Then click OK. Output: One call produced the
5.106 Consider two independent standard normal variables whose joint probability density is 1 2π e −(z2 1 + z2 2)/2 Under a change to polar coordinates, z1 = r cos( θ ),z2 = rsin( θ ), we have r2 = z2 1 + z2 2 and dz1 dz2 = r dr dθ, so the joint density of r and θ is r e−r2/2 1 2π , 0 0
5.105 Verify that (a) the Weibull density α β xβ−1 e−a xβ , x > 0, corresponds to the distribution function F(x) = 1 − e−a xβ , x > 0; (b) the solution of u = F(x) is given by x = − 1 α ln ( 1 − u ) 1/β .
5.104 Verify that (a) the exponential density 0.3 e−0.3x, x > 0 corresponds to the distribution function F(x) = 1 − e−0.3x, x > 0; (b) the solution of u = F(x) is given by x = [− ln ( 1 − u ) ] / 0.3.
5.103 (Transformations) The MINITAB commands Dialog box: Calc > Calculator Type C2 in Store. Type LOGE(C1) in Expression. Click OK. Calc > Calculator Type C3 in Store. Type SQRT(C1) Expression. Click OK. Calc > Calculator Type C4 in Store. Type SQRT(C3) Expression. Click OK. will place ln x in C2,
5.102 (Normal scores plots) The MINITAB commands Dialog box: Calc > Calculator Type C2 in Store. Type NSCOR(C1) in Expression. Click OK. Graph > Scatteplot > Simple. Click OK. Type C1 under Y and C2 under X. Click OK. will create a normal scores plot from observations that were set in C1. (MINITAB
5.101 For any 11 observations, (a) Use software or Table 3 to verify the normal scores −1.38 − 0.97 − 0.67 − 0.43 − 0.21 0 0.21 0.43 0.67 0.97 1.38 (b) Construct a normal scores plot using the observations on the times between neutrinos in Exercise 2.7.
5.100 Let X have the geometric distribution f (x) = p ( 1 − p ) x−1 for x = 1, 2,... (a) Obtain the moment generating function for t < − ln ( 1 − p ) [ Hint: Recall that ∞ k=0 rk = 1 1 − r for |r| < 1. ] (b) Obtain E(X) and E(X2) by differentiating the moment generating function.
5.99 Let X and Y be independent normal random variables with E(X) = 4 and σ2 X = 25 E(Y ) = 3 and σ2 Y = 16 (a) Use moment generating functions to show that 5X − 4Y + 7 has a normal distribution. (b) Find the mean and variance of the random variable in part (a).
5.98 Establish the result in Example 41 concerning the difference of two independent normal random variables, X and Y.
5.97 Let X be a continuous random variable having probability density function f (x) = 2 e−2 x for x > 0 0 elsewhere (a) Find the moment generating function. (b) Obtain E(X) and E(X2) by differentiating the moment generating function.
5.96 Let Z have a normal distribution with mean 0 and variance 1. (a) Find the moment generating function of Z2. (b) Identify the distribution of Z2 by recognizing the form of the moment generating function.
5.95 Let f (x) = 0.40 4 x for x = 0, 1, 2, 3, 4 (a) Find the moment generating function. (b) Obtain E(X) and E(X2) by differentiating the moment generating function.
5.94 Let f (x) = 0.2 for x = 0, 1, 2, 3, 4. (a) Find the moment generating function. (b) Obtain E(X) and E(X2) by differentiating the moment generating function.
5.93 Let X1, X2,..., X20 be independent and let each have the same marginal distribution with mean 10 and variance 3. Find (a) E( X1 + X2 +···+ X20 ); (b) Var ( X1 + X2 +···+ X20 ).
5.92 The time taken by a traditional nuclear reactor to generate one nuclear chain reaction with fast neutrons, X1, has mean 10 nanoseconds and variance 4, while the time taken by an improved design of the reactor, X2, has mean 8 nanoseconds and variance 2.5. Find the expected time savings using
5.91 If X1 has mean 1 and variance 3 while X2 has mean −2 and variance 5, and the two are independent, find (a) E ( X1 + 2X2 − 3 ); (b) Var ( X1 + 2X2 − 3 ).
5.90 If X1 has mean 8 and variance 2 while X2 has mean −12.5 and variance 2.25, and the two are independent, find (a) E( X1 − X2 ); (b) Var ( X1 − X2 ).
5.89 If X1 has mean 1 and variance 5 while X2 has mean −1 and variance 5, and the two are independent, find (a) E( X1 + X2 ); (b) Var ( X1 + X2 ).
5.88 Establish a relationship between f1(x1 | x2), f2(x2 | x1), f1(x1), and f2(x2).
5.87 If measurements of the length and the width of a rectangle have the joint density f (x, y) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 1 a b for L − a 2 < x < L + a 2 , W − b 2 < y < W + b 2 0 elsewhere find the mean and the variance of the corresponding distribution of the area of the
5.86 With reference to Exercise 5.76, find the expected value of the random variable whose values are given by g(x, y) = x2y.
5.85 With reference to Exercise 5.73, find the expected value of the random variable whose values are given by g(x1, x2) = x1 + x2.
5.84 A precision drill positioned over a target point will make an acceptable hole if it is within 5 microns of the target. Using the target as the origin of a rectangular system of coordinates, assume that the coordinates (x, y) of the point of contact are values of a pair of random variables
5.83 A pair of random variables has the circular normal distribution if their joint density is given by f (x1, x2) = 1 2πσ2 e− [ ( x1 − μ1 ) 2 + ( x2 − μ2 ) 2 ] /2σ2 for −∞ < x1 < ∞ and −∞ < x2 < ∞. (a) If μ1 = 2 and μ2 = −2, and σ = 10, use Table 3 to find the
5.82 With reference to the preceding exercise, check whether (a) the three random variables are independent; (b) any two of the three random variables are pairwise independent.
5.81 If three random variables have the joint density f (x, y,z) = ⎧ ⎨ ⎩ k ( x + y ) e−z for 0 < x < 2, 0 < y < 1,z > 0 0 elsewhere find (a) the value of k; (b) the probability that X > Y and Z > 1.
5.80 With reference to Example 27, find expressions for (a) the conditional density of the first random variable when the second takes on the value x2 = 0.25; (b) the conditional density of the second random variable when the first takes on the value x1.
5.79 With reference to Exercise 5.76, find (a) an expression for f1(x | y) for 0 < y < 1; (b) an expression for f1(x | 0.5); (c) the mean of the conditional density of the first random variable when the second takes on the value 0.5.
5.78 With reference to Exercise 5.76, find both marginal densities and use them to find the probabilities that (a) X > 0.8; (b) Y < 0.5.
5.77 With reference to the preceding exercise, find the joint cumulative distribution function of the two random variables and use it to verify the value obtained for the probability.
5.76 If two random variables have the joint density f (x, y) = ⎧ ⎨ ⎩ 6 5 ( x + y2 ) for 0 < x < 1, 0 < y < 1 0 elsewhere find the probability that 0.2 < X < 0.5 and 0.4 < Y < 0.6.
5.75 With reference to Exercise 5.73, find the joint cumulative distribution function of the two random variables, the cumulative distribution functions of the individual random variables, and check whether the two random variables are independent.
5.74 With reference to the preceding exercise, find the marginal densities of the two random variables.
5.73 If two random variables have the joint density f (x1, x2) = x1x2 for 0 < x1 < 2, 0 < x2 < 1 0 elsewhere find the probabilities that (a) both random variables will take on values less than 1; (b) the sum of the values taken on by the two random variables will be less than 1.
5.72 Two random variables are independent and each has a binomial distribution with success probability 0.7 and 4 trials. (a) Find the joint probability distribution. (b) Find the probability that the first random variable is greater than the second.
5.71 Two transistors are needed for an integrated circuit. Of the eight available, three have broken insulation layers, two have poor diodes, and three are in good condition. Two transistors are selected at random. (a) Find the joint probability distribution of X1 = the number of transistors with
5.70 Verify the formula for the variance of the Weibull distribution given on page 160.
5.69 Suppose that the processing speed (in milliseconds) of a supercomputer is a random variable having the Weibull distribution with α = 0.005 and β = 0.125. What is the probability that such a supercomputer will have similar processing speeds after running for 50,000 ms?
5.68 Suppose that the time to failure (in minutes) of certain electronic components subjected to continuous vibrations may be looked upon as a random variable having the Weibull distribution with α = 1 5 and β = 1 3 . (a) How long can such a component be expected to last? (b) What is the
5.67 With reference to the Example 19, find the probability that such a battery will not last 100 hours.
5.65 Suppose the proportion of error in code developed by a programmer, which varies from software to software, may be looked upon as a random variable having the beta distribution with α = 2 and β = 7. (a) Find the mean of this beta distribution, namely, the average proportion of errors in a
5.64 If the ratio of defective switches produced during complete production cycles in the previous month can be looked upon as a random variable having a beta distribution with α = 3 and β = 6, what is the probability that in any given year, there will be fewer than 5% defective switches
5.63 Verify for α = 3 and β = 3 that the integral of the beta density, from 0 to 1, is equal to 1.
5.62 Use the result of Exercise 5.61 to find an expression for the probability density of the waiting time between successive arrivals.
5.61 Given a Poisson process with on the average α arrivals per unit time, find the probability that there will be no arrivals during a time interval of length t, namely, the probability that the waiting times between successive arrivals will be at least of length t.
5.60 With reference to Exercise 4.58, find the probabilities that the time between successive requests for consulting will be (a) less than 0.5 week; (b) more than 3 weeks.
5.59 With reference to Exercise 4.95, find the percent of the time that the interval between breakdowns of the computer will be (a) less than 1 week; (b) at least 5 weeks.
5.58 The server of a multinational corporate network can run for an amount of time without having to be rebooted and this amount of time is a random variable having the exponential distribution β = 30 days. Find the probabilities that such a server will (a) have to be rebooted in less than 10
5.57 Show that when α > 1, the graph of the gamma density has a relative maximum at x = β(α − 1). What happens when 0
5.56 Verify the expression for the variance of the gamma distribution given on page 156.
5.55 With reference to the Example 14, suppose the expert opinion is in error. Calculate the probability that the supports will survive if (a) μ = 3.0 and σ2 = 0.09; (b) μ = 4.0 and σ2 = 0.25.
5.54 At a construction site, the daily requirement of gneiss (in metric tons) is a random variable having a gamma distribution with α = 2 and β = 5. If their supplier’s daily supply capacity is 25 metric tons, what is the probability that this capacity will be inadequate on any given day?
5.53 With reference to Exercise 5.52, find the probabil- ity that the random variable will take on a value less than 5.
5.52 If a random variable has the gamma distribution with a = 2 and =3, find the mean and the standard de- viation of this distribution.
5.51 With reference to the preceding exercise, find the prob- abilities that the random variable will take on a value (a) less than 8.0; (b) between 4.5 and 6.5.
5.50 If a random variable has the log-normal distribution with a = -3 and = 3, find its mean and its stan- dard deviation.
5.49 With reference to the Example 12, find the probability that will take on a value between 7.0 and 7.5.
5.48 Verify the expression given on page 154 for the mean of the log-normal distribution.
5.47 From experience Mr. Harris has found that the low bid on a construction job can be regarded as a random variable having the uniform density f (x) = ⎧ ⎪⎨ ⎪⎩ 3 4C for 2C 3 < x < 2C 0 elsewhere whereC is his own estimate of the cost of the job. What percentage should Mr. Harris add to
5.46 In a manufacturing process, the error made in determining the composition of an alloy is a random variable having the uniform density with α = −0.075 and β = 0.010. What are the probabilities that such an error will be (a) between 0.050 and 0.001? (b) between 0.001 and 0.008?
5.45 Find the distribution function of a random variable having a uniform distribution on (0, 1).
5.44 Normal probabilities can be calculated using MINITAB. Let X have a normal distribution with mean 11.3 and standard deviation 5.7. The following steps yield the cumulative probability of 9.31 or smaller, or P(X ≤ 9.31). Dialog box: Calc > Probability Distribution > Normal Choose Cumulative
5.43 Verify that the parameter σ2 in the expression for the normal density on page 140 is, in fact, its variance.
5.42 Verify that the parameter μ in the expression for the normal density on page 140, is, in fact, its mean.
5.41 Verify the identity F(−z) = 1 − F(z) given on page 141.
5.40 To illustrate the law of large numbers mentioned on Page 116, find the probabilities that the proportion of drawing a club from a fair deck of cards will be anywhere from 0.24 to 0.26 when a card is drawn (a) 100 times; (b) 10,000 times.
5.39 Refer to Example 11 concerning the experiment that confirms electron antineutrinos change type. Suppose instead that there are 400 electron antineutrinos leaving the reactor. Repeat parts (a)–(c) of the example.
5.38 Workers in silicon factories are prone to a lung disease called silicosis. In a recent survey in a factory, about 11% of the workers have been infected by it. Assume the same rate of infection holds everywhere. Use the normal distribution to approximate the probability that, out of a random
5.37 The probability that an electronic component will fail in less than 1,000 hours of continuous use is 0.25. Use the normal approximation to find the probability that among 200 such components fewer than 45 will fail in less than 1,000 hours of continuous use.
5.36 From past experience, a company knows that, on average, 5% of their concrete does not meet standards. Use the normal approximation of the binomial distribution to determine the probability that among 2000 bags of concrete, 75 bags contain concrete that does not meet standards.
5.35 If a random variable has the binomial distribution with n = 25 and p = 0.65, use the normal approximation to determine the probabilities that it will take on (a) the value 15; (b) a value less than 10.
5.34 An automatic machine fills distilled water in 500-ml bottles. Actual volumes are normally distributed about a mean of 500 ml and their standard deviation is 20 ml. (a) What proportion of the bottles are filled with water outside the tolerance limit of 475 ml to 525 ml? (b) To what value does
5.33 The quantity of aerated water that a machine puts in a bottle of a carbonated beverage follows a normal distribution with a standard deviation of 0.25 g. At what “normal” (mean) weight should the machine be set so that no more than 8% of the bottles have more than 20 g of aerated water?
5.32 The number of teeth of a 12% tooth gear produced by a machine follows a normal distribution. Verify that if σ = 1.5 and the mean number of teeth is 13, 74% of the gears contain at least 12 teeth.
5.31 A machine produces soap bars with a weight of 80 ± 0.10 g. If the weight of the soap bars manufactured by the machine may be looked upon as a random variable having normal distribution with μ = 80.05 g and σ = 0.05 g, what percentage of these bars will meet specifications?
5.30 With reference to the preceding exercise, for which temperature is the probability 0.05 that it will be exceeded during one day?
5.29 The daily high temperature in a computer server room at the university can be modeled by a normal distribution with mean 68.7◦F and standard deviation 1.2◦F. Find the probability that, on a given day, the high temperature will be (a) between 68.3 and 70.3◦F (b) greater than 71.5◦F.
5.28 Find the quartiles −z0.25 z0.50 z0.25 of the standard normal distribution.
5.27 Refer to Exercise 5.26 but suppose that a large potential contract contains the specification that at most 7.5% can have a flow coefficient less than 420 Cv. If the manufacturing process is improved to meet this specification, determine (a) the new mean μ if the standard deviation is 25 Cv;
5.26 Butterfly-style valves used in heating and ventilating industries have a high flow coefficient. Flow coefficient can be modeled by a normal distribution with mean 496 Cv and standard deviation 25 Cv. Find the probability that a valve will have a flow coefficient of (a) at least 450 Cv; (b)
5.25 The time for oil to percolate to all parts of an engine can be treated as a random variable having a normal distribution with mean 20 seconds. Find its standard deviation if the probability is 0.25 that it will take a value greater than 31.5 seconds.
5.24 Given a random variable having the normal distribution with μ = 16.2 and σ2 = 1.5625, find the probabilities that it will take on a value (a) greater than 16.8; (b) less than 14.9; (c) between 13.6 and 18.8; (d) between 16.5 and 16.7.
5.23 Verify that (a) z0.005 = 2.575; (b) z0.025 = 1.96.
5.22 If a random variable has a normal distribution, what are the probabilities that it will take on a value within (a) 1 standard deviation of the mean; (b) 2 standard deviations of the mean;(c) 3 standard deviations of the mean; (d) 4 standard deviations of the mean?
5.21 The nozzle of a mixing vibrator is tested for its number of vibrations. The vibration frequency, for each nozzle sample, can be modeled by a normal distribution with mean 128 and standard deviation 16 PdM. (a) If engineering specifications require the sample to have a vibration frequency of
5.20 If a random variable has the standard normal distribution, find the probability that it will take on a value (a) between 0 and 2.3; (b) between 1.22 and 2.43; (c) between −1.45 and −0.45; (d) between −1.70 and 1.35.
5.19 If a random variable has the standard normal distribution, find the probability that it will take on a value (a) less than 1.75; (b) less than −1.25; (c) greater than 2.06; (d) greater than −1.82.

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