As we shall illustrate in Chapter 12, statistical methods

Problem 1E The current in a certain circuit as measured by an ammeter is a continuous random variable ?X ?with the following density function: a.? ?Graph the pdf and verify that the total area under the density curve is indeed 1. b. Calculate P(X ? 4). How does this probability compare to P(X < 4)? c.? ?Calculate P(3.5 ? X ? 4.5) and also P( 4.5 < X).

Problem 2E Suppose the reaction temperature ?X ?(in ?C) in a certain chemical process has a uniform distribution with A = -5 and B = 5. a.? ?Compute P(X <0). b.? ompute P(-2.5 < X <2.5). c.? ?ComputeP(-2 ? X ? 3). d. For k satisfying -5 < k < k + 4 <5, compute P(k < X < k + 4).

Problem 3E The error involved in making a certain measurement is a continuous rv ?X ?with pdf a.?? ketch the graph of ?f (? ? ? . b.? ?Compute P(X > 0). c.? ?Compute P(-1 < X < 1). d.?? omputeP(X <-.5 or X > .5)

Problem 4E Let ?X ?denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article “Blade Fatigue Life Assessment with Application to VAWTS” (?J. of Solar Energy Engr., ?1982: 107–111) proposes the Rayleigh distribution, with pdf as a model for the ?X ?distribution. a.? ?Verify that? ? ?) is a legitimate pdf. b.? ?Suppose ? = 100 (a value suggested by a graph in the article). What is the probability that ?X ? s at most 200? Less than 200? At least 200? c.? ?What is the probability that ?X ?is between 100 and 200 (again assuming ? = 100)? d.?? ive an expression for P(X ? x).

Problem 5E A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X= the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of ?X ?is a.? ?Find the value of ?k ?and draw the corresponding density curve. [?Hint?: Total area under the graph of ?f?(? is 1.] b.? ?What is the probability that the lecture ends within 1 min of the end of the hour? c.? ?What is the probability that the lecture continues beyond the hour for between 60 and 90 sec? d.? ?What is the probability that the lecture continues for at least 90 sec beyond the end of the hour?

Problem 6E The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous rv X ? ?with pdf a.? ketch the graph of ?f(? ? . b.? ?Find the value of ?k?. c.? ?What is the probability that the actual tracking weight is greater than the prescribed weight? d.? ?What is the probability that the actual weight is within .25 g of the prescribed weight? e.? ?What is the probability that the actual weight differs from the prescribed weight by more than .5 g?

Problem 7E The article ?"Second Moment Reliability Evaluation vs. Monte Carlo Simulations for Weld Fatigue Strength" (Quality and Reliability Engr. Intl., 2012: 887496) ?considered the use of a uniform distribution with A = .20 and B = 4.2.5 for the diameter X of a certain type of weld (mm). a. Determine the pdf of X and graph it. b. What is the probability that diameter exceeds 3 mm? c. What is the probability that diameter is within I mm of the mean diameter? d. For any value a satisfying .20 < a < a + 1 < 4.25, what is P(a < X < a + 1)?

Problem 8E In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with A = 0and B = 0 , then it can be shown that the total waiting time Y? ?has the pdf a.?? ketch a graph of the pdf of ?Y?. b.?? erify that c.?? hat is the probability that total waiting time is at most 3 min? d.?? hat is the probability that total waiting time is at most 8 min? e.?? hat is the probability that total waiting time is between 3 and 8 min? f.? ?What is the probability that total waiting time is either less than 2 min or more than 6 min?

Problem 10E A family of pdf’s that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, ?k ?and , both >0, and the pdf is a.? ?Sketch the graph of f ? ? ? ?k?, ?). b.? ?Verify that the total area under the graph equals 1. c.? ?If the rv ?X ?has pdf? f ?(?x?; ?k?, ?), for any fixed b > ? obtain an expression forP(X?b). d.? ?For ?P(?a ??? ?? ?b? .

Problem 11E Let ?X ?denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is Use the cdf to obtain the following:

Problem 12E The cdf for ? ? = measurement error) of Exercise 3 is a. Compute P(x<0). b. Compute P(X-1 c. Compute P(.5 d. Verify that? ?x?) is as given in Exercise 3 by obtainingF’(x). e. Verify that

Problem 13E Example 4.5 introduced the concept of time headway in traffic flow and proposed a particular distribution for X = the headway between two randomly selected consecutive cars (sec). Suppose that in a different traffic environment, the distribution of time headway has the form a.? etermine the value of ?k ?for which ?f?(?x?) is a legitimate pdf. b.? ?Obtain the cumulative distribution function. c.? ?Use the cdf from (b) to determine the probability that headway exceeds 2 sec and also the probability that headway is between 2 and 3 sec. d.? ?Obtain the mean value of headway and the standard deviation of headway. e.? ?What is the probability that headway is within 1 standard deviation of the mean value?

Problem 14E The article “Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants” (?Water Research, ?1984: 1169–1174) suggests the uniform distribution on the interval (7.5, 20) as a model for depth (cm) of the bioturbation layer in sediment in a certain region. a.? ?What are the mean and variance of depth? b.? hat is the cdf of depth? c.? ?What is the probability that observed depth is at most 10? Between 10 and 15? d.? ?What is the probability that the observed depth is within 1 standard deviation of the mean value? Within 2 standard deviations?

Problem 15E Let ? ?denote the amount of space occupied by an article placed in a 1-1ft3 packing container. The pdf of ?X i? s a. ? raph the pdf. Then obtain the cdf of X? ? nd graph it. b. ?What is P(X?.5) c. ?Using the cdf from (a), what is P(.25 < X ? .5)? What is P(.25 ? X ? .5)? d. ? hat is the 75th percentile of the distribution? e. ?Compute E? ? ? ? and ? x . f. ?What is the probability that ?X ?is more than 1 standard deviation from its mean value?

Problem 16E

Problem 17E

Problem 18E Let ?X ?denote the voltage at the output of a microphone, and suppose that ?X ?has a uniform distribution on the interval from to –1. The voltage is processed by a “hard limiter” with cutoff values and – .5, so the limiter output is a random variableY? ?related to ?X ?by Y = X if |X| ? .5 Y = .5 if X> .5, andY = – .5 if X<– .5 . a. What is P( Y = .5)? b. Obtain the cumulative distribution function ofY? ?and graph it.

Let X be a continuous rv with cdf \(F(x)=\left\{\begin{array}{cl} 0 & x \leq 0 \\ \frac{x}{4}\left[1+\ln \left(\frac{4}{x}\right)\right] & 0<x \leq 4 \\ 1 & x>4 \end{array}\right.\) [This type of cdf is suggested in the article “Variability in Measured Bedload-Transport Rates” (Water Resources Bull., 1985: 39–48) as a model for a certain hydrologic variable.] What is a. \(P(X\ \leq\ 1)\)? b. \(P((1\ \leq\ X\ \leq\ 3)\)? c.The pdf of X?

Problem 20E Consider the pdf for total waiting time Y? ? or two buses introduced in Exercise 8. a.? ?Compute and sketch the cdf of ?Y?. [?Hint?: Consider separately 0?y<5 and 5?y?10 in computing ?F?(?y?). A graph of the pdf should be helpful.] b.? ?Obtain an expression for the (100?p?)th percentile. [?Hint?: Consider separately 0 c.? ?Compute ?E?(?Y ?) and ?V?(?Y?). How do these compare with the expected waiting time and variance for a single bus when the time is uniformly distributed on [0, 5]? Reference exercise 8 In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with A = 0 and B = 5 , then it can be shown that the total waiting time Y? ?has the pdf a.?? ketch a graph of the pdf of ?Y?. b.?? erify that . c.?? hat is the probability that total waiting time is at most 3 min? d.?? hat is the probability that total waiting time is at most 8 min? e.?? hat is the probability that total waiting time is between 3 and 8 min? f.? ?What is the probability that total waiting time is either less than 2 min or more than 6 min?

An ecologist wishes to mark off a circular sampling region having radius 10 m. However, the radius of the resulting region is actually a random variable R with the following pdf. f(r) = 3/4 \(1 - (14 - r)^2\) 0 \(13\ \leq\ r\ \leq\ 15\) What is the expected area of the resulting circular region?

Problem 22E The weekly demand for propane gas (in 1000s of gallons) from a particular facility is an rv ?X ?with pdf a.? ?Compute the cdf of ?X?. b.? ?Obtain an expression for the (100?p?)th percentile. What is the value of ? c.? ?Compute ?E?? ?) and ?? ?? . d.? ?If 1.5 thousand gallons are in stock at the beginning of the week and no new supply is due in during the week, how much of the 1.5 thousand gallons is expected to be left at the end of the week? [?Hint?: h(x)Let amount left when demand = ?x.? ]

Problem 23E If the temperature at which a certain compound melts is a random variable with mean value 120 °and standard deviation 2 °C, what are the mean temperature and standard deviation measured in ?F[Hint : 1.8 °C + 32.]

Let X have the Pareto pdf \(f(x, k, \theta)=\left\{\begin{array}{cl} \frac{k \cdot \theta^{k}}{x^{k+1}} & x \geq \theta \\ 0 & x<\theta \end{array}\right.\) introduced in Exercise 10. a. If k > 1, compute E(X). b. What can you say about E(X) if k = 1? c. If k > 2 , show that \(V(X) = k \theta^2 (k – 1)^{–2} (k –2)^{–1}\). d. If k = 2, what can you say about V(X)? e. What conditions on k are necessary to ensure that \(E(X^n)\) is finite?

Problem 25E Let ?X ?be the temperature in °C at which a certain chemical reaction takes place, and let ?Y ?be the temperature in °F(so Y = 1.8X + 32) a.?? f the median of the ?X ?distribution is X + 32)is the median of the ?Y distribution. b.? ?How is the 90th percentile of the ?Y ?distribution related to the 90th percentile of the ? ? istribution? Verify your conjecture. c.?? ore generally, if Y = aX +b , how is any particular percentile of the ?Y distribution related to the corresponding percentile of the ?X ?distribution?

Problem 26E Let ?X ?be the total medical expenses (in 1000s of dollars)incurred by a particular individual during a given year. Although ?X ?is a discrete random variable, suppose its distribution is quite well approximated by a continuous distribution with pdf f(x) = k ( 1+ x/2.5)–7for x?0 . a.? hat is the value of ?k?? b.? ?Graph the pdf of ?X?. c.? ?What are the expected value and standard deviation of total medical expenses? d.? ?This individual is covered by an insurance plan that entails a $500 deductible provision (so the first $500 worth of expenses are paid by the individual). Then the plan will pay 80% of any additional expenses exceeding $500, and the maximum payment by the individual (including the deductible amount) is $2500. Let ?Y ?denote the amount of this individual’s medical expenses paid by the insurance company. What is the expected value of ?Y?? [?Hint?: First figure out what value of ?X ?corresponds to the maximum out-of-pocket expense of $2500. Then write an expression for ?Y ?as a function of ?X ?(which involves several different pieces) and calculate the expected value of this function.]

Problem 27E When a dart is thrown at a circular target, consider the location of the landing point relative to the bull’s eye. Let ?X ?be the angle in degrees measured from the horizontal, and assume that ?X ?is uniformly distributed on [0, 360]. Define ?Y ?to be the transformed variable Y = h(X) = ( 2?/360)X – ?, so ?Y ?is the angle measured in radians and ?Y ?is between -? and . Obtain ?E?(?Y?) and ?Y by first obtaining ?E?(?X?) and ?x, and then using the fact that h ? ? ?X?) is a linear function of ?X?.

Problem 28E Let ? ?be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate.

Problem 29E In each case, determine the value of the constant ?c ?that makes the probability statement correct.

Problem 30E Find the following percentiles for the standard normal distribution. Interpolate where appropriate. a.? ?91st b.?? th c.? ?75th d.? ?25th e.?? th

Problem 31E Determine? z?? for the following: a.? = .0055 b.? = .09 c.? = .663

Problem 32E Suppose the force acting on a column that helps to support a building is a normally distributed random variable ?X ?with mean value 15.0 kips and standard deviation 1.25 kips. Compute the following probabilities by standardizing and then using Table A.3. a. P(X ? 15) b. P(X ? 17.5) c. P(X ? 10) d. P(14 ? X ? 18) e. P(|X – 15 | ? 3 )

Problem 33E Mopeds (small motorcycles with an engine capacity below50cm3 ) are very popular in Europe because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections” (?J. of Automobile Engr., ?2008: 1615–1623) described a rolling bench test for determining maximum vehicle speed. A normal distribution with mean value 46.8 km/h and standard deviation 1.75 km/h is postulated. Consider randomly selecting a single such moped. a.? hat is the probability that maximum speed is at most 50 km/h? b.? hat is the probability that maximum speed is at least 48 km/h? c.? ?What is the probability that maximum speed differs from the mean value by at most 1.5 standard deviations?

Problem 34E The article “Reliability of Domestic-Waste Biofilm Reactors” (?J. of Envir. Engr., 1995: 785–790) suggests that substrate concentration (mg/cm3 )of influent to a reactor is normally distributed with ? = .30and ?= .06. a.? hat is the probability that the concentration exceeds .25? b.? hat is the probability that the concentration is at most .10? c.? ?How would you characterize the largest 5% of all concentration values?

In a road-paving process, asphalt mix is delivered to the hopper of the paver by trucks that haul the material from the batching plant. The article "Modeling of Simultaneously Continuous and Stochastic Construction Activities for Simulation" U. of Construction Engr. and Mgmnt., 2013: 1037-1045) proposed a normal distribution with mean value 8.46 min and standard deviation .913 min for the rv X = truck haul time. a. What is the probability that haul time will be at least 10 min? Will exceed 10 min? b. What is the probability that haul time will exceed 15 min? c. What is the probability that haul time will be between 8 and 10 min? d. What value c is such that 98% of all haul times are in the interval from 8.46 — c to 8.4.6 + c? e. If four haul times are independently selected, what is the probability that at least one of them exceeds 10 min?

Problem 36E Spray drift is a constant concern for pesticide applicators and agricultural producers. The inverse relationship between droplet size and drift potential is well known. The paper “Effects of 2,4-D Formulation and Quinclorac on Spray Droplet Size and Deposition” (?Weed Technology, ?2005: 1030–1036) investigated the effects of herbicide formulation on spray atomization. A figure in the paper suggested the normal distribution with mean 1050 ?m and standard deviation 150 ?m was a reasonable model for droplet size for water (the “control treatment”) sprayed through a 760 ml/min nozzle. a.? ?What is the probability that the size of a single droplet is less than 1500 ?m ? At least 1000?m ? b.? ?What is the probability that the size of a single droplet is between 1000 ?m and ? c.? ow would you characterize the smallest 2% of all droplets? d.? ?If the sizes of five independently selected droplets are measured, what is the probability that at least one exceeds 1500?m?

Problem 37E Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 104 and standard deviation 5 (information in the article “Mathematical Model of Chloride Concentration in Human Blood,” ?J. of Med. Engr. and Tech?., 2006: 25–30, including a normal probability plot as described in Section 4.6, supports this assumption). a.? ?What is the probability that chloride concentration equals 105? Is less than 105? Is at most 105? b.? ?What is the probability that chloride concentration differs from the mean by more than 1 standard deviation? Does this probability depend on the values of and ? c.? ?How would you characterize the most extreme .1% of chloride concentration values?

Problem 38E There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean 3 cm and standard deviation .1 cm. The second machine produces corks with diameters that have a normal distribution with mean 3.04 cm and standard deviation .02 cm. Acceptable corks have diameters between 2.9 cm and 3.1 cm. Which machine is more likely to produce an acceptable cork?

Problem 39E The defect length of a corrosion defect in a pressurized steel pipe is normally distributed with mean value 30 mm and standard deviation 7.8 mm [suggested in the article ?"Reliability Evaluation of Corroding Pipelines Considering Multiple Failure Modes and Time-Dependent Internal Pressure" („I. of Infrastructure Systems, 2011: 216-224)]. a. What is the probability that defect length is at most 20 mm? Less than 20 mm? b. What is the 75th percentile of the defect length distribution—that is, the value that separates the smallest 75% of all lengths from the largest 25%? c. What is the 15th percentile of the defect length distribution? d. What values separate the middle 80% of the defect length distribution from the smallest 10% and the largest 10%?

Problem 40E The article “Monte Carlo Simulation—Tool for Better Understanding of LRFD” (?J. of Structural Engr., ?1993: 1586–1599) suggests that yield strength (ksi) for A36 grade steel is normally distributed with ? = 43 and ? = 4.5. a.? ?What is the probability that yield strength is at most 40? Greater than 60? b.? ?What yield strength value separates the strongest 75% from the others?

Problem 41E The automatic opening device of a military cargo parachute has been designed to open when the parachute is 200 m above the ground. Suppose opening altitude actually has a normal distribution with mean value 200 m and standard deviation 30 m. Equipment damage will occur if the parachute opens at an altitude of less than 100 m. What is the probability that there is equipment damage to the payload of at least one of five independently dropped parachutes?

Problem 42E The temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean , the actual temperature of the medium, and standard deviation . What would the value of ? have to be to ensure that 95% of all readings are within. 1 ° of ??

Problem 43E Vehicle speed on a particular bridge in China can k modeled as normally distributed ?("Fatigue Reliability Assessment for Long-Span Bridges under Combined Dynamic Loads from Winds and Vehicles," J. of Bridge Engr., 2013: 735-747). a. If 5% of all vehicles travel less than 39.12 milt and • 10% travel more than 73.24 m/h, what are the mean and standard deviation of vehicle speed? [Note: The resulting values should agree with those given in the cited article.) b. What is the probability that a randomly selected vehicle's speed is between 50 and 65 m/h? c. What is the probability that a randomly selected vehicle's speed exceeds the speed limit of 70 m/h?

Problem 44E If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is a.? ithin 1.5 SDs of its mean value? b.? ?Farther than 2.5 SDs from its mean value? c.? ?Between 1 and 2 SDs from its mean value?

A machine that produces ball bearings has initially been set so that the true average diameter of the bearings it produces is .500 in. A bearing is acceptable if its diameter is within .004 in. of this target value. Suppose, however, that the setting has changed during the course of production, so that the bearings have normally distributed diameters with mean value .499 in. and standard deviation .002 in. What percentage of the bearings produced will not be acceptable?

Problem 47E The weight distribution of parcels sent in a certain manner is normal with mean value 12 lb and standard deviation 3.5 lb. The parcel service wishes to establish a weight value ?c ?beyond which there will be a surcharge. What value of ?c ?is such that 99% of all parcels are at least 1 lb under the surcharge weight?

Problem 48E Suppose Appendix Table A.3 contained ?(z) only for z ?0 Explain how you could still compute a. P( –1.72? Z ?–.55) b. P( –1.72? Z ? .55) Is it necessary to tabulate ?(z) for ?z ?negative? What property of the standard normal curve justifies your answer?

Problem 49E Consider babies born in the “normal” range of 37–43 weeks gestational age. Extensive data supports the assumption that for such babies born in the United States, birth weight is normally distributed with mean 3432 g and standard deviation 482 g. [The article “Are Babies Normal?” (?The American Statistician, 1999: 298–302) analyzed data from a particular year; for a sensible choice of class intervals, a histogram did not look at all normal, but after further investigations it was determined that this was due to some hospitals measuring weight in grams and others measuring to the nearest ounce and then converting to grams. A modified choice of class intervals that allowed for this gave a histogram that was well described by a normal distribution.] a.? ?What is the probability that the birth weight of a randomly selected baby of this type exceeds 4000 g? Is between 3000 and 4000 g? b.? ?What is the probability that the birth weight of a randomly selected baby of this type is either less than 2000 g or greater than 5000 g? c.? ?What is the probability that the birth weight of a randomly selected baby of this type exceeds 7 lb? d.? ow would you characterize the most extreme .1% of all birth weights? e.? ?If ?X ?is a random variable with a normal distribution and ?a ?is a numerical constant (a ?0), then Y = aX also has a normal distribution. Use this to determine the distribution of birth weight expressed in pounds (shape, mean, and standard deviation), and then recalculate the probability from part (c). How does this compare to your previous answer?

Problem 51E Chebyshev’s inequality, (see Exercise 44, Chapter 3), is valid for continuous as well as discrete distributions. It states that for any number k satisfying k ? 1,P(|X – ? ?k?) ?1/k2 (see Exercise 44 in Chapter 3 for an interpretation). Obtain this probability in the case of a normal distribution for , 2, and 3, and compare to the upper bound. Reference exercise 44 A result called ?Chebyshev’s inequality ?states that for any probability distribution of an rv ?X ?and any number ?k ?that is at least 1,? P?( | ?X ?- µ | k ?) ? 1/?k?2 . In words, the probability that the value of ?X ?lies at least ?k ?standard deviations from its mean is at most 1/?k?2. a.? hat is the value of the upper bound for k = 2? K + 3? K = 4? K= 5? K = 10? b.? ?Compute µ and ? for the distribution of Exercise 13. Then evaluate P(|X - µ| ? k?) for the values of ?k ?given in part (a). What does this suggest about the upper bound relative to the corresponding probability? c.? ?Let ?X ?have possible values -1, 0, and 1, with probabilities 1/18, 8/9 and 1/8 , respectively. What is P(|X - µ|? 3?), and how does it compare to the corresponding bound? d.? ?Give a distribution for which P(|X - µ|? 5?) = .04.

Problem 57E a.? ?Show that if ?X ?has a normal distribution with parameters ? and ? , then Y = aX + b (a linear function of ?X?) also has a normal distribution. What are the parameters of the distribution of ?Y ?[i.e., ?E?(?Y?) and ?V?(?Y?)]? [?Hint?: Write the cdf of Y?,P(Y ?y) , as an integral involving the pdf of ?X?, and then differentiate with respect to ?y? o get the pdf of ?Y.? ] b.? ?If, when measured in °C , temperature is normally distributed with mean 115 and standard deviation 2, what can be said about the distribution of temperature measured in ° F?

Problem 65E Let X denote the data transfer time(ms) in a grid computing system (the time required for data transfer between a "worker" computer and a "master" computer. Suppose that X has a gamma distribution with mean value 37.5 ins and standard deviation 21.6 (suggested by the article ?"Computation Time of Grid Computing with Data Transfer Times that Follow a Gamma Distribution," Proceedings of the First International Conference on Semantics, Knowledge, and Grid, 2005). a. What are the values of a and p? b. What is the probability that data transfer time exceeds 50 ms? c. What is the probability that data transfer time is between 50 and 75 ms?

Problem 70E If ?X ?has an exponential distribution with parameter , derive a general expression for the (100?p?)th percentile of the distribution. Then specialize to obtain the median.

Problem 72E The lifetime ?X ?(in hundreds of hours) of a certain type of vacuum tube has a Weibull distribution with parameters ? and ? . Compute the following: a.?? ? ?X?) an? ?(?X?) b. P(X ? 6) c.P(1.5 ? X ? 6) (This Weibull distribution is suggested as a model for time in service in “On the Assessment of Equipment Reliability: Trading Data Collection Costs for Precision,” ?J. of Engr. Manuf., ?1991: 105–109.)

Problem 75E Let ?X ?have a Weibull distribution with the pdf from Expression (4.11). Verify that ? =?+(1+ 1/?). [?Hint?: In the integral for ?E?(?X?)?, ?make the change of variable y = (x/?)? ,so that x = ?y1/?.]

Problem 80E a.? se Equation (4.13) to write a formula for the median of the lognormal distribution. What is the median for the load distribution of Exercise 79? b.? ?Recalling that z? 100(1 – ?)is our notation for the percentile of the standard normal distribution, write an expression for the 100(1 – ?) percentile of the lognormal distribution. In Exercise 79, what value will load exceed only 1% of the time? Reference exercise 79 Nonpoint source loads are chemical masses that travel to the main stem of a river and its tributaries in flows that are distributed over relatively long stream reaches, in contrast to those that enter at well-defined and regulated points. The article “Assessing Uncertainty in Mass Balance Calculation of River Nonpoint Source Loads” (?J. of Envir. Engr., 2 ? 008: 247–258) suggested that for a certain time period and location X = nonpoint source load of total dissolved solids could be modeled with a lognormal distribution having mean value 10,281 kg/day/km and a coefficient of variation CV = .40(cv = ?x/?x). a.?? hat are the mean value and standard deviation of ln(?X) ? ? b.?? hat is the probability that ?X ?is at most 15,000 kg/day/km? c.? ?What is the probability that ?X ?exceeds its mean value, and why is this probability not .5? d.? ?Is 17,000 the 95th percentile of the distribution? Reference equation 4.13

Problem 86E Stress is applied to a 20-in. steel bar that is clamped in a fixed position at each end. Let Y = the distance from the left end at which the bar snaps. Suppose ?Y?/20 has a standard beta distribution with E(Y) = 10 and V(Y) = 100/7 a.? hat are the parameters of the relevant standard beta distribution? b.? ?Compute P(8 ? Y ? 12). c.? ?Compute the probability that the bar snaps more than 2 in. from where you expect it to.

Problem 94E The accompanying observations are precipitation values during March over a 30-year period in Minneapolis-St. Paul. a.? onstruct and interpret a normal probability plot for this data set. b.? ?Calculate the square root of each value and then construct a normal probability plot based on this transformed data. Does it seem plausible that the square root of precipitation is normally distributed? c.? ?Repeat part (b) after transforming by cube roots.

Problem 117E Let Z have a standard normal distribution and define a new rv ?Y ?by Y = ? Z + ? . Show that ?Y ?has a normal distribution with parameters ? and ? . [?Hint?: Y ? y iff Z ? ? Use this to find the cdf of ?Y ?and then differentiate it with respect to ?y ?.]

The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given in Exercise 3.13 on page 92 as x 01234 f(x) 0.41 0.37 0.16 0.05 0.01 Find the average number of imperfections per 10 meters of this fabric.

The probability distribution of the discrete random variable X is f(x)=3 x 1 4x3 43x ,x=0 ,1,2,3. Find the mean of X.

Find the mean of the random variable T representing the total of the three coins in Exercise 3.25 on page 93.

A coin is biased such that a head is three times as likely to occur as a tail. Find the expected number of tails when this coin is tossed twice.

In a gambling game, a woman is paid $3 if she draws a jack or a queen and $5 if she draws a king or an ace from an ordinary deck of 52 playing cards. If she draws any other card, she loses. How much should she pay to play if the game is fair?

An attendant at a car wash is paid according to the number of cars that pass through. Suppose the probabilities are 1/12, 1/12, 1/4, 1/4, 1/6, and 1/6, respectively, that the attendant receives $7, $9, $11, $13, $15, or $17 between 4:00 P.M. and 5:00 P.M. on any sunny Friday. Find the attendants expected earnings for this particular period.

By investing in a particular stock, a person can make a prot in one year of $4000 with probability 0.3 or take a loss of $1000 with probability 0.7. What is this persons expected gain?

Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are 0.22, 0.36, 0.28, and 0.14, respectively, that she will be able to sell it for a prot of $250, sell it for a prot of $150, break even, or sell it for a loss of $150. What is her expected prot?

A private pilot wishes to insure his airplane for $200,000. The insurance company estimates that a total loss will occur with probability 0.002, a 50% loss with probability 0.01, and a 25% loss with probability 0.1. Ignoring all other partial losses, what premium should the insurance company charge each year to realize an average prot of $500?

Two tire-quality experts examine stacks of tires and assign a quality rating to each tire on a 3-point scale. Let X denote the rating given by expert A and Y denote the rating given by B. The following table gives the joint distribution for X and Y . y f(x,y) 123 1 0.10 0.05 0.02 x 2 0.10 0.35 0.05 3 0.03 0.10 0.20 Find X and Y .

The density function of coded measurements of the pitch diameter of threads of a tting is f(x)= 4 (1+x2), 0 <x<1, 0, elsewhere. Find the expected value of X.

If a dealer’s profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function \(f(x)=\left\{\begin{array}{ll} 2(1-x), & 0<x<1 \\ 0, & \text { elsewhere } \end{array}\right.\) find the average profit per automobile.

The density function of the continuous random variable X, the total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year, is given in Exercise 3.7 on page 92 as \(f(x)=\left\{\begin{array}{ll} x, & 0<x<1 \\ 2-x, & 1 \leq x<2 \\ 0, & \text { elsewhere } \end{array}\right.\) Find the average number of hours per year that families run their vacuum cleaners.

Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function f(x)=2(x+2) 5 , 0 <x<1, 0, elsewhere.

Assume that two random variables (X, Y ) are uniformly distributed on a circle with radius a. Then the joint probability density function is \(f(x, y)=\left\{\begin{array}{ll} \frac{1}{\pi a^{2}}, & x^{2}+y^{2} \leq a^{2}, \\ 0, & \text { otherwise, } \end{array}\right.\) Find \(\mu_X\), the expected value of X.

Suppose that you are inspecting a lot of 1000 light bulbs, among which 20 are defectives. You choose two light bulbs randomly from the lot without replacement. Let X1 =1, if the 1st light bulb is defective, 0, otherwise, X2 =1, if the 2nd light bulb is defective, 0, otherwise. Find the probability that at least one light bulb chosen is defective. [Hint: Compute P(X1 + X2 = 1).]

Let X be a random variable with the following probability distribution: x 3 6 9f (x) 1/6 1/2 1/3 Find g(X), where g(X) = (2X + 1) 2.

Find the expected value of the random variable g(X)=X2, where X has the probability distribution of Exercise 4.2.

A large industrial rm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous year. Suppose that the number of word processors, X, purchased each year has the following probability distribution: x 0 1 2 3 f(x) 1/10 3/10 2/5 1/5 If the cost of the desired model is $1200 per unit and at the end of the year a refund of 50X2 dollars will be issued, how much can this rm expect to spend on new word processors during this year?

A continuous random variable X has the density function f(x)=ex, x > 0, 0, elsewhere. Find the expected value of g(X)=e2X/3.

What is the dealers average prot per automobile if the prot on each automobile is given by g(X)=X2, where X is a random variable having the density function of Exercise 4.12?

The hospitalization period, in days, for patients following treatment for a certain type of kidney disorder is a random variable Y = X + 4, where X has the density function f(x)= 32 (x+4)3 , x > 0, 0, elsewhere. Find the average number of days that a person is hospitalized following treatment for this disorder.

Suppose that X and Y have the following joint probability function: x f(x,y) 24 1 0.10 0.15 y 3 0.20 0.30 5 0.10 0.15 (a) Find the expected value of g(X,Y )=XY2. (b) Find X and Y .

Referring to the random variables whose joint probability distribution is given in Exercise 3.39 on page 105, (a) nd E(X2Y 2XY); (b) nd X Y .

Referring to the random variables whose joint probability distribution is given in Exercise 3.51 on page 106, nd the mean for the total number of jacks and kings when 3 cards are drawn without replacement from the 12 face cards of an ordinary deck of 52 playing cards.

Let X and Y be random variables with joint density function f(x,y)=4xy, 0 < x, y <1, 0, elsewhere. Find the expected value of Z = X2 + Y 2.

In Exercise 3.27 on page 93, a density function is given for the time to failure of an important component of a DVD player. Find the mean number of hours to failure of the component and thus the DVD player.

Consider the information in Exercise 3.28 on page 93. The problem deals with the weight in ounces of the product in a cereal box, with f(x)=2 5, 23.75 x 26.25, (a) Plot the density function. (b) Compute the expected value, or mean weight, in ounces. (c) Are you surprised at your answer in (b)? Explain why or why not.

Exercise 3.29 on page 93 dealt with an important particle size distribution characterized by f(x)=3x4, x > 1, 0, elsewhere. (a) Plot the density function. (b) Give the mean particle size.

In Exercise 3.31 on page 94, the distribution of times before a major repair of a washing machine was given as \(f(y)=\left\{\begin{array}{ll} \frac{1}{4} e^{-y / 4}, & y \geq 0 \\ 0, & \text { elsewhere } \end{array}\right.\) What is the population mean of the times to repair?

Consider Exercise 3.32 on page 94. (a) What is the mean proportion of the budget allocated to environmental and pollution control? (b) What is the probability that a company selected at random will have allocated to environmental and pollution control a proportion that exceeds the population mean given in (a)?

In Exercise 3.13 on page 92, the distribution of the number of imperfections per 10 meters of synthetic fabric is given by x 01234 f(x) 0.41 0.37 0.16 0.05 0.01 (a) Plot the probability function. (b) Find the expected number of imperfections, E(X)=. (c) Find E(X2).

Use Denition 4.3 on page 120 to nd the variance of the random variable X of Exercise 4.7 on page 117.

Let X be a random variable with the following probability distribution: Find the standard deviation of X.

The random variable X, representing the number of errors per 100 lines of software code, has the following probability distribution: Using Theorem 4.2 on page 121, find the variance of X.

Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1, respectively, that 0, 1, 2, or 3 power failures will strike a certain subdivision in any given year. Find the mean and variance of the random variable X representing the number of power failures striking this subdivision.

A dealers prot, in units of $5000, on a new automobile is a random variable X having the density function given in Exercise 4.12 on page 117. Find the variance of X.

The proportion of people who respond to a certain mail-order solicitation is a random variable X having the density function given in Exercise 4.14 on page 117. Find the variance of X.

The total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a random variable X having the density function given in Exercise 4.13 on page 117. Find the variance of X.

Referring to Exercise 4.14 on page 117, nd 2 g(X) for the function g(X)=3X2 + 4.

Find the standard deviation of the random variable g(X) = (2X + 1) 2 in Exercise 4.17 on page 118.

Using the results of Exercise 4.21 on page 118, nd the variance of g(X)=X2, where X is a random variable having the density function given in Exercise 4.12 on page 117.

The length of time, in minutes, for an airplane to obtain clearance for takeo at a certain airport is a random variable Y =3X2, where X has the density function f(x)=1 4ex/4, x > 0 0, elsewhere. Find the mean and variance of the random variable Y .

Find the covariance of the random variables X and Y of Exercise 3.39 on page 105.

Find the covariance of the random variables X and Y of Exercise 3.49 on page 106.

Find the covariance of the random variables X and Y of Exercise 3.44 on page 105.

For the random variables X and Y whose joint density function is given in Exercise 3.40 on page 105, find the covariance.

Given a random variable X, with standard deviation \(\sigma_X\), and a random variable Y = a + bX, show that if b < 0, the correlation coefficient \(\rho_{XY} = ?1\), and if b > 0, \(\rho_{XY} = 1\).

Consider the situation in Exercise 4.32 on page 119. The distribution of the number of imperfections per 10 meters of synthetic failure is given by x 01234 f(x) 0.41 0.37 0.16 0.05 0.01 Find the variance and standard deviation of the number of imperfections.

For a laboratory assignment, if the equipment is working, the density function of the observed outcome X is \(f(x)=\left\{\begin{array}{ll} 2(1-x), & 0<x<1, \\ 0, & \text { otherwise, } \end{array}\right.\) Find the variance and standard deviation of X.

For the random variables X and Y in Exercise 3.39 on page 105, determine the correlation coecient between X and Y .

Random variables X and Y follow a joint distribution f(x,y)=2, 0 <x y<1, 0, otherwise. Determine the correlation coecient between X and Y .

Referring to Exercise 4.35 on page 127, nd the mean and variance of the discrete random variable Z =3X 2, when X represents the number of errors per 100 lines of code.

Using Theorem 4.5 and Corollary 4.6, nd the mean and variance of the random variable Z =5X+3, where X has the probability distribution of Exercise 4.36 on page 127.

Suppose that a grocery store purchases 5 cartons of skim milk at the wholesale price of $1.20 per carton and retails the milk at $1.65 per carton. After the expiration date, the unsold milk is removed from the shelf and the grocer receives a credit from the dis tributor equal to three-fourths of the wholesale price. If the probability distribution of the random variable X, the number of cartons that are sold from this lot, is x 012345 f(x) 1 15 2 15 2 15 3 15 4 15 3 15 nd the expected prot.

Repeat Exercise 4.43 on page 127 by applying Theorem 4.5 and Corollary 4.6.

Let X be a random variable with the following probability distribution: Find E(X) and \(E(X^2)\) and then, using these values, evaluate \(E[(2X + 1)^2\)].

The total time, measured in units of 100 hours, that a teenager runs her hair dryer over a period of one year is a continuous random variable X that has the density function f(x)= x, 0 <x<1, 2x, 1 x<2, 0, elsewhere. Use Theorem 4.6 to evaluate the mean of the random variable Y = 60 X2 + 39X, where Y is equal to the number of kilowatt hours expended annually

If a random variable X is dened such that E[(X 1)2] = 10 and E[(X 2)2]=6 , nd and 2.

Suppose that X and Y are independent random variables having the joint probability distribution x f(x,y) 24 1 0.10 0.15 y 3 0.20 0.30 5 0.10 0.15 Find (a) E(2X 3Y ); (b) E(XY).

Use Theorem 4.7 to evaluate E(2XY2 X2Y ) for the joint probability distribution shown in Table 3.1 on page 96.

If X and Y are independent random variables with variances 2 X = 5 and 2 Y = 3, nd the variance of the random variable Z = 2X +4Y 3.

Repeat Exercise 4.62 if X and Y are not independent and XY = 1.

Suppose that X and Y are independent random variables with probability densities and g(x)=8 x3 , x > 2, 0, elsewhere, and h(y)=2y, 0 <y<1, 0, elsewhere. Find the expected value of Z = XY.

Let X represent the number that occurs when a red die is tossed and Y the number that occurs when a green die is tossed. Find (a) E(X + Y ); (b) E(X Y ); (c) E(XY).

Let X represent the number that occurs when a green die is tossed and Y the number that occurs when a red die is tossed. Find the variance of the random variable (a) 2X Y ; (b) X +3Y 5.

If the joint density function of X and Y is given by \(f(x, y)=\left\{\begin{array}{ll} \frac{2}{7}(x+2 y), & 0<x<1,1<y<2 \\ 0, & \text { elsewhere } \end{array}\right.\) find the expected value of \(g(X, Y) = \frac{X}{Y^3}+X^2 Y\).

The power P in watts which is dissipated in an electric circuit with resistance R is known to be given by P = I2R, where I is current in amperes and R is a constant xed at 50 ohms. However, I is a random variable with I = 15 amperes and 2 I =0 .03 amperes2. Give numerical approximations to the mean and variance of the power P.

Consider Review Exercise 3.77 on page 108. The random variables X and Y represent the number of vehicles that arrive at two separate street corners during a certain 2-minute period in the day. The joint distribution is f(x,y)= 1 4(x+y)9 16, for x =0 ,1,2,... and y =0 ,1,2,.... (a) Give E(X), E(Y ), Var(X), and Var(Y ). (b) Consider Z = X + Y , the sum of the two. Find E(Z) and Var(Z).

Consider Review Exercise 3.64 on page 107. There are two service lines. The random variables X and Y are the proportions of time that line 1 and line 2 are in use, respectively. The joint probability density function for (X,Y ) is given by f(x,y)=3 2(x2 + y2), 0 x, y 1, 0, elsewhere. (a) Determine whether or not X and Y are independent. (b) It is of interest to know something about the proportion of Z = X + Y , the sum of the two proportions. Find E(X + Y ). Also nd E(XY). (c) Find Var(X), Var(Y ), and Cov(X,Y ). (d) Find Var(X + Y ).

The length of time Y , in minutes, required to generate a human reex to tear gas has the density function f(y)=1 4ey/4, 0 y<, 0, elsewhere. (a) What is the mean time to reex? (b) Find E(Y 2) and Var(Y ).

A manufacturing company has developed a machine for cleaning carpet that is fuel-ecient because it delivers carpet cleaner so rapidly. Of interest is a random variable Y , the amount in gallons per minute delivered. It is known that the density function is given by f(y)=1, 7 y 8, 0, elsewhere. (a) Sketch the density function. (b) Give E(Y ), E(Y 2), and Var(Y ).

For the situation in Exercise 4.72, compute \(E(e^Y)\) using Theorem 4.1, that is, by using \(E\left(e^{Y}\right)=\int_{7}^{8} e^{y} f(y) d y\) Then compute \(E(e^Y)\) not by using f(y), but rather by using the second-order adjustment to the first-order approximation of \(E(e^Y)\). Comment.

Consider again the situation of Exercise 4.72. It is required to find \(Var(e^Y)\). Use Theorems 4.2 and 4.3 and define \(Z = e^Y\). Thus, use the conditions of Exercise 4.73 to find \(\operatorname{Var}(Z)=E\left(Z^{2}\right)-[E(Z)]^{2}\) Then do it not by using f(y), but rather by using the first-order Taylor series approximation to \(Var(e^Y)\). Comment!

An electrical rm manufactures a 100-watt light bulb, which, according to specications written on the package, has a mean life of 900 hours with a standard deviation of 50 hours. At most, what percentage of the bulbs fail to last even 700 hours? Assume that the distribution is symmetric about the mean.

Seventy new jobs are opening up at an automobile manufacturing plant, and 1000 applicants show up for the 70 positions. To select the best 70 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathematical ability. The mean grade on this test turns out to be 60, and the scores have a standard deviation of 6. Can a person who scores 84 count on getting one of the jobs? [Hint: Use Chebyshevs theorem.] Assume that the distribution is symmetric about the mean.

A random variable X has a mean = 10 and a variance 2 = 4. Using Chebyshevs theorem, nd (a) P(|X 10|3); (b) P(|X 10| < 3); (c) P(5 <X<15); (d) the value of the constant c such that P(|X 10|c) 0.04.

Compute P( 2<X<+2), where X has the density function

Prove Chebyshevs theorem.

Find the covariance of random variables X and Y having the joint probability density function f(x,y)=x + y, 0 <x<1, 0 <y<1, 0, elsewhere.

Referring to the random variables whose joint probability density function is given in Exercise 3.47 on page 105, nd the average amount of kerosene left in the tank at the end of the day.

Assume the length X, in minutes, of a particular type of telephone conversation is a random variable with probability density function f(x)=1 5ex/5, x > 0, 0, elsewhere. (a) Determine the mean length E(X) of this type of telephone conversation. (b) Find the variance and standard deviation of X. (c) Find E[(X + 5) 2].

Referring to the random variables whose joint density function is given in Exercise 3.41 on page 105, nd the covariance between the weight of the creams and the weight of the toees in these boxes of chocolates.

Referring to the random variables whose joint probability density function is given in Exercise 3.41 on page 105, nd the expected weight for the sum of the creams and toees if one purchased a box of these chocolates.

Suppose it is known that the life X of a particular compressor, in hours, has the density function f(x)= 1 900ex/900, x > 0, 0, elsewhere. (a) Find the mean life of the compressor. (b) Find E(X2). (c) Find the variance and standard deviation of the random variable X.

Referring to the random variables whose joint density function is given in Exercise 3.40 on page 105, (a) find \(\mu_X\) and \(\mu_Y\) ; (b) find E[(X + Y )/2].

Show that Cov(aX,bY )=ab Cov(X,Y ).

Consider the density function of Review Exercise 4.85. Demonstrate that Chebyshevs theorem holds for k = 2 and k = 3.

Consider the joint density function f(x,y)=16y x3 , x > 2, 0 <y<1, 0, elsewhere. Compute the correlation coecient XY .

Consider random variables X and Y of Exercise 4.63 on page 138. Compute XY .

A dealers prot, in units of $5000, on a new automobile is a random variable X having density function f(x)=2(1x), 0 x 1, 0, elsewhere. (a) Find the variance of the dealers prot. (b) Demonstrate that Chebyshevs theorem holds for k = 2 with the density function above. (c) What is the probability that the prot exceeds $500?

Consider Exercise 4.10 on page 117. Can it be said that the ratings given by the two experts are independent? Explain why or why not.

A companys marketing and accounting departments have determined that if the company markets its newly developed product, the contribution of the product to the rms prot during the next 6 months will be described by the following: Prot Contribution Probability $5,000 $10,000 $30,000 0.2 0.5 0.3 What is the companys expected prot?

In a support system in the U.S. space program, a single crucial component works only 85% of the time. In order to enhance the reliability of the system, it is decided that 3 components will be installed in parallel such that the system fails only if they all fail. Assume the components act independently and that they are equivalent in the sense that all 3 of them have an 85% success rate. Consider the random variable X as the number of components out of 3 that fail. (a) Write out a probability function for the random variable X. (b) What is E(X) (i.e., the mean number of components out of 3 that fail)? (c) What is Var(X)? (d) What is the probability that the entire system is successful? (e) What is the probability that the system fails? (f) If the desire is to have the system be successful with probability 0.99, are three components sucient? If not, how many are required?

In business, it is important to plan and carry out research in order to anticipate what will occur at the end of the year. Research suggests that the profit (loss) spectrum for a certain company, with corresponding probabilities, is as follows: (a) What is the expected profit? (b) Give the standard deviation of the profit.

It is known through data collection and considerable research that the amount of time in seconds that a certain employee of a company is late for work is a random variable X with density function f(x)= 3 (4)(503)(502 x2), 50 x 50, 0, elsewhere. In other words, he not only is slightly late at times, but also can be early to work. (a) Find the expected value of the time in seconds that he is late. (b) Find E(X2). (c) What is the standard deviation of the amount of time he is late?

A delivery truck travels from point A to point B and back using the same route each day. There are four traffic lights on the route. Let \(X_1\) denote the number of red lights the truck encounters going from A to B and \(X_2\) denote the number encountered on the return trip. Data collected over a long period suggest that the joint probability distribution for (\(X_1,\ X_2\)) is given by (a) Give the marginal density of \(X_1\). (b) Give the marginal density of \(X_2\). (c) Give the conditional density distribution of \(X_1\) given \(X_2 = 3\). (d) Give \(E(X_1)\). (e) Give \(E(X_2)\). (f) Give \(E(X_1\ |\ X_2 = 3)\). (g) Give the standard deviation of \(X_1\).

A convenience store has two separate locations where customers can be checked out as they leave. These locations each have two cash registers and two employees who check out customers. Let X be the number of cash registers being used at a particular time for location 1 and Y the number being used at the same time for location 2. The joint probability function is given by y x 012 0 0.12 0.04 0.04 1 0.08 0.19 0.05 2 0.06 0.12 0.30 (a) Give the marginal density of both X and Y as well as the probability distribution of X given Y = 2. (b) Give E(X) and Var(X). (c) Give E(X | Y = 2) and Var(X | Y = 2).

Consider a ferry that can carry both buses and cars across a waterway. Each trip costs the owner approximately $10. The fee for cars is $3 and the fee for buses is $8. Let X and Y denote the number of buses and cars, respectively, carried on a given trip. The joint distribution of X and Y is given by x y 012 0 0.01 0.01 0.03 1 0.03 0.08 0.07 2 0.03 0.06 0.06 3 0.07 0.07 0.13 4 0.12 0.04 0.03 5 0.08 0.06 0.02 Compute the expected prot for the ferry trip.

As we shall illustrate in Chapter 12, statistical methods associated with linear and nonlinear models are very important. In fact, exponential functions are often used in a wide variety of scientic and engineering problems. Consider a model that is t to a set of data involving measured values k1 and k2 and a certain response Y to the measurements. The model postulated is Y = eb0+b1k1+b2k2, where Y denotes the estimated value of Y, k1 and k2 are xed values, and b0,b1, and b2 are estimates of constants and hence are random variables. Assume that these random variables are independent and use the approximate formula for the variance of a nonlinear function of more than one variable. Give an expression for Var( Y ). Assume that the means of b0,b1, and b2 are known and are 0, 1, and 2, and assume that the variances of b0,b1, and b2 are known and are 2 0, 2 1, and 2 2.

Consider Review Exercise 3.73 on page 108. It involved Y , the proportion of impurities in a batch, and the density function is given by f(y)=10(1y)9, 0 y 1, 0, elsewhere. (a) Find the expected percentage of impurities. (b) Find the expected value of the proportion of quality material (i.e., nd E(1Y )).(c) Find the variance of the random variable Z =1Y .

Project: Let X = number of hours each student in the class slept the night before. Create a discrete variable by using the following arbitrary intervals: X<3, 3 X<6, 6 X<9, and X 9. (a) Estimate the probability distribution for X. (b) Calculate the estimated mean and variance for X.