Applied Statistics and Probability for Engineers 6th Edition solutions

Applied Statistics and Probability for Engineers 6
Determine the mean and variance of a beta random variable. Use the result that the probability density function integrates to 1. That is, \(\frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha+\beta)}=\int_{0}^{1} x^{\alpha-1}(1-x)^{\beta-1}\) for \(\alpha>0, \beta>0\).

Applied Statistics and Probability for Engineers 6
The two-parameter exponential distribution uses a different range for the random variable \(\text { X }\), namely, \(0 \leq y \leq x\) for a constant ? (and this equals the usual exponential distribution in the special case that \(y=0\)). The probability density function for \(\text { X }\)is f \((x

Applied Statistics and Probability for Engineers 6
Problem 234MEE 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 upper and lower specifications at a distance of six standard deviati

Applied Statistics and Probability for Engineers 6
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?

Applied Statistics and Probability for Engineers 6
Let the random variable \(X\) denote a measurement from a manufactured product. Suppose that 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

Applied Statistics and Probability for Engineers 6
Problem 230MEE 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 will fail before 60,000 hours?

Applied Statistics and Probability for Engineers 6
Lack of Memory Property. Show that for an exponential random variable \(X, P\left(X<t_{1}+t_{2} \mid X>t_{1}\right)=P\left(X<t_{2}\right)\). Equation Transcription: Text Transcription: X,P(X<t_1+t_2|X>t_1)=P(X<t_2)

Applied Statistics and Probability for Engineers 6
Suppose that X has a beta distribution with parameters \(\alpha = 2.5\) and \(\beta = 2.5\). Sketch an approximate graph of the probability density function. Is the density symmetric?

Applied Statistics and Probability for Engineers 6
Suppose that x has a beta distribution with parameters \(\alpha=2.5\) and \(\beta=1\). Determine the following: (a) \(P(X<0.25)\) (b) \(P(0.25<X<0.75)\) (c) Mean and variance Equation transcription: Text transcription: \alpha=2.5 \beta=1 P(X<0.25) P(0.25<X<0.75)

Applied Statistics and Probability for Engineers 6
Suppose that X has a beta distribution with parameters \(\alpha=1\) and \(\beta=4.2\). Determine the following: (a) \(P(X<0.25)\) (b) \(P(0.5<X)\) (c) Mean and variance Equation transcription: Text transcription: \alpha=1 \beta=4.2 P(X<0.25) P(0.5<X)

Applied Statistics and Probability for Engineers 6
A European standard value for a low-emission window glazing uses 0.59 as the proportion of solar energy that enters a room. Suppose that the distribution of the proportion of solar energy that enters a room is a beta random variable. (a) Calculate the mode, mean, and variance of the distribution

Applied Statistics and Probability for Engineers 6
The length of stay at a hospital emergency department is the sum of the waiting and service times. Let X denote the proportion of time spent waiting and assume a beta distribution with \(\alpha=10\) and \(\beta=1\). Determine the following: (a) \(P(X>0.9)\) (b) \(P(X<0.5)\)

Applied Statistics and Probability for Engineers 6
The maximum time to complete a task in a project is 2.5 days. Suppose that the completion time as a proportion of this maximum is a beta random variable with \(\alpha=2\) and \(\beta=3\)\). What is the probability that the task requires more than two days to complete?

Applied Statistics and Probability for Engineers 6
An allele is an alternate form of a gene, and the proportion of alleles in a population is of interest in genetics. An article in BMC Genetics [Calculating Expected DNA Remnants From Ancient Founding Events in Human Population Genetics (2008, Vol. 9:66)] used a beta distribution with mean 0.3 and sta

Applied Statistics and Probability for Engineers 6
Suppose that the construction of a solar power station is initiated. The projects completion time has not been set due to uncertainties in financial resources. The completion time for the first phase is modeled with a beta distribution and the minimum, most likely (mode), and maximum completion times

Applied Statistics and Probability for Engineers 6
The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample is \(f(x)=0.04\) for \(50<X<75\) seconds. (a) What percentage of tests requires more than 70 seconds to complete? (b) What percentage of tests requires less than one minute to complete

Applied Statistics and Probability for Engineers 6
The tensile strength of paper is modeled by a normal distribution with a mean of 35 pounds per square inch and a standard deviation of 2 pounds per square inch. (a) What is the probability that the strength of a sample is less than ? (b) If the specifications require the tensile strength to exceed

Applied Statistics and Probability for Engineers 6
The time it takes a cell to divide (called mitosis) is normally distributed with an average time of one hour and a standard deviation of five minutes. (a) What is the probability that a cell divides in less than 45 minutes? (b) What is the probability that it takes a cell more than 65 minutes to di

Applied Statistics and Probability for Engineers 6
The length of an injection-molded plastic case that holds magnetic tape is normally distributed with a length of 90.2 millimeters and a standard deviation of 0.1 millimeter. (a) What is the probability that a part is longer than 90.3 millimeters or shorter than 89.7 millimeters? (b) What should the p

Applied Statistics and Probability for Engineers 6
Problem 196SE The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours. (a) What is the probability that the sick-leave time for next month will be between 50 and 80 hours? (b) How much time should be budgeted for sick

Applied Statistics and Probability for Engineers 6
Problem 197SE The percentage of people exposed to a bacteria who become ill is 20%. Assume that people are independent. Assume that 1000 people are exposed to the bacteria. Approximate each of the following: (a) Probability that more than 225 become ill (b) Probability that between 175 and 225 become

Applied Statistics and Probability for Engineers 6
The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with \(\lambda=0.00004\). What is the probability that the time until failure is (a) At least 20,000 hours? (b) At most 30,000 hours? (c) Between 20,000 and 30,000 hours? Eq

Applied Statistics and Probability for Engineers 6
Problem 199SE When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each caller orders one ticket. What

Applied Statistics and Probability for Engineers 6
Problem 200SE The time between process problems in a manufacturing line is exponentially distributed with a mean of 30 days. (a) What is the expected time until the fourth problem? (b) What is the probability that the time until the fourth problem exceeds 120 days?

Applied Statistics and Probability for Engineers 6
The life of a recirculating pump follows a Weibull distribution with parameters \(\beta=2\) and \(\delta=700\) hours. Determine for parts (a) and (b): (a) Mean life of a pump (b) Variance of the life of a pump (c) What is the probability that a pump will last longer than its m

Applied Statistics and Probability for Engineers 6
Problem 202SE The size of silver particles in a photographic emulsion is known to have a log normal distribution with a mean of 0.001 mm and a standard deviation of 0.002 mm. (a) Determine the parameter values for the lognormal distribution. (b) What is the probability of a particle size greater than

Applied Statistics and Probability for Engineers 6
Suppose that \(f(x)=0.5 x-1\) for \(2<x<4\). Determine the following: (a) \(P(X<2.5)\) (b) \(P(X>3)\) (c) \(P(2.5<X<3.5)\) (d) Determine the cumulative distribution function of the random variable. (e) Determine the mean and variance of the random variable. Equat

Applied Statistics and Probability for Engineers 6
Problem 204SE The time between calls is exponentially distributed with a mean time between calls of 10 minutes. (a) What is the probability that the time until the first call is less than five minutes? (b) What is the probability that the time until the first call is between 5 and 15 minutes? (c) Det

Applied Statistics and Probability for Engineers 6
Problem 205SE The CPU of a personal computer has a lifetime that is exponentially distributed with a mean lifetime of six years. You have owned this CPU for three years. (a) What is the probability that the CPU fails in the next three years? (b) Assume that your corporation has owned 10 CPUs for thre

Applied Statistics and Probability for Engineers 6
Suppose that X has a lognormal distribution with parameters \(\theta=0\) and \(\omega^{2}=4\). Determine the following: (a) \(P(10<X<50)\) (b) Value for x such that \(P(X<x)=0.05\) (c) Mean and variance of X Equation transcription: Text transcription: \theta=0 \omega^{2}=4 P(10<X<50) P(X<

Applied Statistics and Probability for Engineers 6
Suppose that X has a lognormal distribution and that the mean and variance of X are 50 and 4000, respectively. Determine the following: (a) Parameters \(\theta\) and \(\omega^{2}\) of the lognormal distribution (b) Probability that X is less than 150 Equation transcription: Text transcription

Applied Statistics and Probability for Engineers 6
Problem 208SE Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per square centimeter of surface dust is 100. A sample of 800 square centimeters of dust is an

Applied Statistics and Probability for Engineers 6
Problem 209SE Without an automated irrigation system, the height of plants two weeks after germination is normally distributed with a mean of 2.5 centimeters and a standard deviation of 0.5 centimeter. (a) What is the probability that a plants height is greater than 2.25 centimeters? (b) What is the

Applied Statistics and Probability for Engineers 6
Problem 210SE With an automated irrigation system, a plant grows to a height of 3.5 centimeters two weeks after germination. Without an automated system, the height is normally distributed with mean and standard deviation 2.5 and 0.5 centimeters, respectively. (a) What is the probability of obtaining

Applied Statistics and Probability for Engineers 6
The thickness of a laminated covering for a wood surface is normally distributed with a mean of five millimeters and a standard deviation of 0.2 millimeter. (a) What is the probability that a covering thickness is more than 5.5 millimeters? (b) If the specifications require the thickness to b

Applied Statistics and Probability for Engineers 6
Problem 212SE The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch. (a) Suppose that the specifications require the dot diameter to be between 0.0014 and 0.0026 inch. If the probability that a dot meets specifications is to be 0.9973, what standard

Applied Statistics and Probability for Engineers 6
The waiting time for service at a hospital emergency department follows an exponential distribution with a mean of three hours. Determine the following: (a) Waiting time is greater than four hours (b) Waiting time is greater than six hours given that you have already waited two hours (c) Value x (in

Applied Statistics and Probability for Engineers 6
Problem 214SE The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours. (a) What is the probability that a laser fails before 5800 hours? (b) What is the life in hours that 90% of the lasers exceed? (c) What should t

Applied Statistics and Probability for Engineers 6
Problem 215SE Continuation of Exercise 4-214. Rework parts (a) and (b). Assume that the lifetime is an exponential random variable with the same mean. 4-214.The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours. (

Applied Statistics and Probability for Engineers 6
Continuation of Exercise 4-214. Rework parts (a) and (b). Assume that the lifetime is a lognormal random variable with the same mean and standard deviation.

Applied Statistics and Probability for Engineers 6
Problem 217SE A square inch of carpeting contains 50 carpet fibers. The probability of a damaged fiber is 0.0001. Assume that the damaged fibers occur independently. (a) Approximate the probability of one or more damaged fibers in one square yard of carpeting. (b) Approximate the probability of four

Applied Statistics and Probability for Engineers 6
Problem 218SE 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 who arrive can be seated. (b) Approximate t

Applied Statistics and Probability for Engineers 6
Suppose that the construction of a solar power station is initiated. The projects completion time has not been set due to uncertainties in financial resources. The proportion of completion within one year has a beta distribution with parameters \(\alpha=1\) and \(\beta=5\). Determine the following: (

Applied Statistics and Probability for Engineers 6
An article in IEEE Journal on Selected Areas in Communications [Impulse Response Modeling of Indoor Radio Propagation Channels (1993, Vol. 11(7), pp. 967978)] indicated that the successful design of indoor communication systems requires characterization of radio propagation. The distribution of the a

Applied Statistics and Probability for Engineers 6
Consider the regional right ventricle transverse wall motion in patients with pulmonary hypertension (PH). The right ventricle ejection fraction (EF) is approximately normally distributed with standard deviation of 12 for PH subjects, and with mean and standard deviation of 56 and 8, respectively, fo

Applied Statistics and Probability for Engineers 6
Provide approximate sketches for beta probability density functions with the following parameters. Comment on any symmetries and show any peaks in the probability density functions in the sketches. (a) \(\alpha=\beta<1\) (b) \(\alpha=\beta=1\). (

Applied Statistics and Probability for Engineers 6
Problem 223SE Among homeowners in a metropolitan area, 25% recycle paper each week. A waste management company services 10,000 homeowners (assumed independent). Approximate the following probabilities: (a) More than 2600 recycle paper in a week (b) Between 2400 and 2600 recycle paper in a week (c) Nu

Applied Statistics and Probability for Engineers 6
An article in Journal of Theoretical Biology [Computer Model of Growth Cone Behavior and Neuronal Morphogenesis (1995, Vol. 174(4), pp. 381389)] developed a model for neuronal morphogenesis in which neuronal growth cones have a significant function in the development of the nervous system. This model

Applied Statistics and Probability for Engineers 6
An article in Electric Power Systems Research [On the Self-Scheduling of a Power Producer in Uncertain Trading Environments' ' (2008, Vol. 78(3), pp. 311317)] considered a self scheduling approach for a power producer. In addition to price and forced outages, another uncertainty was due to generation

Applied Statistics and Probability for Engineers 6
Problem 226SE An article in Electronic Journal of Applied Statistical Analysis [Survival Analysis of Acute Myocardial Infarction Patients Using Non-Parametric and Parametric Approaches (2009, Vol. 2(1), pp. 2236)] described the use of a Weibull distribution to model the survival time of acute myocard

Applied Statistics and Probability for Engineers 6
The steps in this exercise lead to the probability density function of an Erlang random variable X with parameters \(\lambda\) and r, \(f(x)=\lambda^{\gamma} X^{\gamma-1} e^{\lambda x} /(r-1) !, x>0\), and \(r=1,2, \ldots\) (a) Use the Poisson distribution to express \(P X x(>)\). (b) Use the result