Chapter 4

The weight of a sophisticated running shoe is normally distributed with a mean of 12 ounces and a standard deviation of 0.5 ounce. (a) What is the probability that a shoe weighs more than 13 ounces? (b) What must the standard deviation of weight be in order for the company to state that \(99.9 \%\) of its shoes weighs less than 13 ounces? (c) If the standard deviation remains at 0.5 ounce, what must the mean weight be for the company to state that \(99.9 \%\) of its shoes weighs less than 13 ounces?

Measurement error that is normally distributed with a mean of 0 and a standard deviation of 0.5 gram is added to the true weight of a sample. Then the measurement is rounded to the nearest gram. Suppose that the true weight of a sample is 165.5 grams. (a) What is the probability that the rounded result is 167 grams? (b) What is the probability that the rounded result is 167 grams or more?

Assume that a random variable is normally distributed with a mean of 24 and a standard deviation of 2 . Consider an interval of length one unit that starts at the value \(a\) so that the interval is \([a, a+1]\). For what value of \(a\) is the probability of the interval greatest? Does the standard deviation affect that choice of interval?

A study by Bechtel et al., \(2009,\) described in the Archives of Environmental \& Occupational Health considered polycyclic aromatic hydrocarbons and immune system function in beef cattle. Some cattle were near major oil- and gas- producing areas of western Canada. The mean monthly exposure to PM1.0 (particulate matter that is \(<1 \mu \mathrm{m}\) in diameter) was approximately 7\. \(1 \mu \mathrm{g} / \mathrm{m}^{3}\) with standard deviation \(1.5 .\) Assume that the monthly exposure is normally distributed. (a) What is the probability of a monthly exposure greater than \(9 \mu g / m^{3} ?\) (b) What is the probability of a monthly exposure between 3 and \(8 \mu \mathrm{g} / \mathrm{m}^{3} ?\) (c) What is the monthly exposure level that is exceeded with probability \(0.05 ?\) (d) What value of mean monthly exposure is needed so that the probability of a monthly exposure more than \(9 \mu \mathrm{g} / \mathrm{m}^{3}\) is \(0.01 ?\)

An article in Atmospheric Chemistry and Physics "Relationship Between Particulate Matter and Childhood AsthmaBasis of a Future Warning System for Central Phoenix" (2012, Vol. \(12,\) pp. \(2479-2490)]\) reported the use of PM10 (particulate matter \(<10 \mu \mathrm{m}\) diameter ) air quality data measured hourly from sensors in Phoenix, Arizona. The 24 -hour (daily) mean PM10 for a centrally located sensor was \(50.9 \mu \mathrm{g} / \mathrm{m}^{3}\) with a standard deviation of \(25.0 .\) Assume that the daily mean of \(\mathrm{PM} 10\) is normally distributed. (a) What is the probability of a daily mean of PM10 greater than \(100 \mu \mathrm{g} / \mathrm{m}^{3} ?\) (b) What is the probability of a daily mean of PM10 less than \(25 \mu \mathrm{g} / \mathrm{m}^{3} ?\) (c) What daily mean of PM10 value is exceeded with probability \(5 \% ?\)

The length of stay at a specific emergency department in Phoenix, Arizona, in 2009 had a mean of 4.6 hours with a standard deviation of \(2.9 .\) Assume that the length of stay is normally distributed. (a) What is the probability of a length of stay greater than 10 hours? (b) What length of stay is exceeded by \(25 \%\) of the visits? (c) From the normally distributed model, what is the probability of a length of stay less than 0 hours? Comment on the normally distributed assumption in this example.

A signal in a communication channel is detected when the voltage is higher than 1.5 volts in absolute value. Assume that the voltage is normally distributed with a mean of \(0 .\) What is the standard deviation of voltage such that the probability of a false signal is \(0.005 ?\)

An article in Microelectronics Reliability ["Advanced Electronic Prognostics through System Telemetry and Pattern Recognition Methods" (2007, Vol.47(12), pp. \(1865-1873\) ) ] presented an example of electronic prognosis. The objective was to detect faults to decrease the system downtime and the number of unplanned repairs in high-reliability systems. Previous measurements of the power supply indicated that the signal is normally distributed with a mean of \(1.5 \mathrm{~V}\) and a standard deviation of \(0.02 \mathrm{~V}\). (a) Suppose that lower and upper limits of the predetermined specifications are \(1.45 \mathrm{~V}\) and \(1.55 \mathrm{~V},\) respectively. What is the probability that a signal is within these specifications? (b) What is the signal value that is exceeded with \(95 \%\) probability? (c) What is the probability that a signal value exceeds the mean by two or more standard deviations?

An article in International Journal of Electrical Power \& Energy Systems ["Stochastic Optimal Load Flow Using a Combined Quasi-Newton and Conjugate Gradient Technique" (1989, Vol.11(2), pp. 85-93)] considered the problem of optimal power flow in electric power systems and included the effects of uncertain variables in the problem formulation. The method treats the system power demand as a normal random variable with 0 mean and unit variance. (a) What is the power demand value exceeded with \(95 \%\) probability? (b) What is the probability that the power demand is positive? (c) What is the probability that the power demand is more than -1 and less than \(1 ?\)

An article in the Journal of Cardiovascular Magnetic Resonance ["Right Ventricular Ejection Fraction Is Better Reflected by Transverse Rather Than Longitudinal Wall Motion in Pulmonary Hypertension" (2010, Vol.12(35)] discussed a study of the regional right ventricle transverse wall motion in patients with pulmonary hypertension (PH). The right ventricle ejection fraction (EF) was approximately normally distributed with a mean and a standard deviation of 36 and \(12,\) respectively, for PH subjects, and with mean and standard deviation of 56 and \(8,\) respectively, for control subjects. (a) What is the EF for PH subjects exceeded with \(5 \%\) probability? (b) What is the probability that the EF of a control subject is less than the value in part (a)? (c) Comment on how well the control and PH subjects can be distinguished by EF measurements.

Suppose that \(X\) is a binomial random variable with \(n=200\) and \(p=0.4\). Approximate the following probabilities: (a) \(P(X \leq 70)\) (b) \(P(70

Suppose that \(X\) is a Poisson random variable with \(\lambda=6\) (a) Compute the exact probability that \(X\) is less than four. (b) Approximate the probability that \(X\) is less than four and compare to the result in part (a). (c) Approximate the probability that \(8

Suppose that \(X\) has a Poisson distribution with a mean of \(64 .\) Approximate the following probabilities: (a) \(P(X>72)\) (b) \(P(X<64)\) (c) \(P(60

The manufacturing of semiconductor chips produces \(2 \%\) defective chips. Assume that the chips are independent and that a lot contains 1000 chips. Approximate the following probabilities: (a) More than 25 chips are defective. (b) Between 20 and 30 chips are defective.

There were 49.7 million people with some type of long-lasting condition or disability living in the United States in \(2000 .\) This represented 19.3 percent of the majority of civilians aged five and over (http://factfinder.census.gov). A sample of 1000 persons is selected at random. (a) Approximate the probability that more than 200 persons in the sample have a disability. (b) Approximate the probability that between 180 and 300 people in the sample have a disability.

Phoenix water is provided to approximately 1.4 million people who are served through more than 362,000 accounts (http:// phoenix.gov/WATER/wtrfacts.html). All accounts are metered and billed monthly. The probability that an account has an error in a month is \(0.001,\) and accounts can be assumed to be independent. (a) What are the mean and standard deviation of the number of account errors each month? (b) Approximate the probability of fewer than 350 errors in a month. (c) Approximate a value so that the probability that the number of errors exceeds this value is \(0.05 .\) (d) Approximate the probability of more than 400 errors per month in the next two months. Assume that results between months are independent.

An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is \(0.999,\) and assume that the components fail independently. Approximate the probability that 10 or more of the original 5000 components fail during the useful life of the product.

A corporate Web site contains errors on 50 of 1000 pages. If 100 pages are sampled randomly without replacement, approximate the probability that at least one of the pages in error is in the sample.

Suppose that the number of asbestos particles in a sample of 1 squared centimeter of dust is a Poisson random variable with a mean of \(1000 .\) What is the probability that 10 squared centimeters of dust contains more than 10,000 particles?

A high-volume printer produces minor print-quality errors on a test pattern of 1000 pages of text according to a Poisson distribution with a mean of 0.4 per page. (a) Why are the numbers of errors on each page independent random variables? (b) What is the mean number of pages with errors (one or more)? (c) Approximate the probability that more than 350 pages contain errors (one or more).

An acticle in Biometrics ["Integrative Analysis of Transcriptomic and Proteomic Data of Desulfovibrio Vulgaris: A Nonlinear Model to Predict Abundance of Undetected Proteins" (2009)\(]\) reported that protein abundance from an operon (a set of biologically related genes) was less dispersed than from randomly selected genes. In the research, 1000 sets of genes were randomly constructed, and of these sets, \(75 \%\) were more disperse than a specific opteron. If the probability that a random set is more disperse than this opteron is truly 0.5 , approximate the probability that 750 or more random sets exceed the opteron. From this result, what do you conclude about the dispersion in the opteron versus random genes?

An article in Atmospheric Chemistry and Physics ["Relationship Between Particulate Matter and Childhood Asthma - Basis of a Future Warning System for Central Phoenix," 2012 , Vol. \(12,\) pp. \(2479-2490]\) linked air quality to childhood asthma incidents. The study region in central Phoenix, Arizona recorded 10,500 asthma incidents in children in a 21 -month period. Assume that the number of asthma incidents follows a Poisson distribution. (a) Approximate the probability of more than 550 asthma incidents in a month. (b) Approximate the probability of 450 to 550 asthma incidents in a month. (c) Approximate the number of asthma incidents exceeded with probability \(5 \%\). (d) If the number of asthma incidents was greater during the winter than the summer, what would this imply about the Poisson distribution assumption?

A set of 200 independent patients take antiacid medication at the start of symptoms, and 80 experience moderate to substantial relief within 90 minutes. Historically, \(30 \%\) of patients experience relief within 90 minutes with no medication. If the medication has no effect, approximate the probability that 80 or more patients experience relief of symptoms. What can you conclude about the effectiveness of this medication?

Among homeowners in a metropolitan area, \(75 \%\) recycle plastic bottles each week. A waste management company services 1500 homeowners (assumed independent). Approximate the following probabilities: (a) At least 1150 recycle plastic bottles in a week (b) Between 1075 and 1175 recycle plastic bottles in a week

Cabs pass your workplace according to a Poisson process with a mean of five cabs per hour. (a) Determine the mean and standard deviation of the number of cabs per 10 -hour day. (b) Approximate the probability that more than 65 cabs pass within a 10 -hour day. (c) Approximate the probability that between 50 and 65 cabs pass in a 10 -hour day. (d) Determine the mean hourly rate so that the probability is approximately 0.95 that 100 or more cabs pass in a 10-hour data.

The number of (large) inclusions in cast iron follows a Poisson distribution with a mean of 2.5 per cubic millimeter. Approximate the following probabilities: (a) Determine the mean and standard deviation of the number of inclusions in a cubic centimeter \((\mathrm{cc})\). (b) Approximate the probability that fewer than 2600 inclusions occur in a cc. (c) Approximate the probability that more than 2400 inclusions occur in a cc. (d) Determine the mean number of inclusions per cubic millimeter such that the probability is approximately 0.9 that 500 or fewer inclusions occur in a cc.

Suppose that \(X\) has an exponential distribution with \(\lambda=2 .\) Determine the following: (a) \(P(X \leq 0)\) (b) \(P(X \geq 2)\) (c) \(P(X \leq 1)\) (d) \(P(1

Suppose that \(X\) has an exponential distribution with mean equal to \(10 .\) Determine the following: (a) \(P(X>10)\) (b) \(P(X>20)\) (c) \(P(X<30)\) (d) Find the value of \(x\) such that \(P(X

Suppose that \(X\) has an exponential distribution with a mean of \(10 .\) Determine the following: (a) \(P(X<5)\) (b) \(P(X<15 \mid X>10)\) (c) Compare the results in parts (a) and (b) and comment on the role of the lack of memory property.

Suppose that the counts recorded by a Geiger counter follow a Poisson process with an average of two counts per minute. (a) What is the probability that there are no counts in a 30 -second interval? (b) What is the probability that the first count occurs in less than 10 seconds? (c) What is the probability that the first count occurs between one and two minutes after start-up?

Suppose that the log-ons to a computer network follow a Poisson process with an average of three counts per minute. (a) What is the mean time between counts? (b) What is the standard deviation of the time between counts? (c) Determine \(x\) such that the probability that at least one count occurs before time \(x\) minutes is \(0.95 .\)

The time between calls to a plumbing supply business is exponentially distributed with a mean time between calls of 15 minutes. (a) What is the probability that there are no calls within a 30-minute interval? (b) What is the probability that at least one call arrives within a 10 -minute interval? (c) What is the probability that the first call arrives within 5 and 10 minutes after opening? (d) Determine the length of an interval of time such that the probability of at least one call in the interval is \(0.90 .\)

The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase a six-year-old automobile, with a working voltage regulator and plan to own it for six years. (a) What is the probability that the voltage regulator fails during your ownership? (b) If your regulator fails after you own the automobile three years and it is replaced, what is the mean time until the next failure?

Suppose that the time to failure (in hours) of fans in a personal computer can be modeled by an exponential distribution with \(\lambda=0.0003 .\) (a) What proportion of the fans will last at least 10,000 hours? (b) What proportion of the fans will last at most 7000 hours?

The time between the arrival of electronic messages at your computer is exponentially distributed with a mean of two hours. (a) What is the probability that you do not receive a message during a two- hour period? (b) If you have not had a message in the last four hours, what is the probability that you do not receive a message in the next two hours? (c) What is the expected time between your fifth and sixth messages?

The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes. (a) What is the probability that you wait longer than one hour for a taxi? (b) Suppose that you have already been waiting for one hour for a taxi. What is the probability that one arrives within the next 10 minutes? (c) Determine \(x\) such that the probability that you wait more than \(x\) minutes is \(0.10 .\) (d) Determine \(x\) such that the probability that you wait less than \(x\) minutes is 0.90 . (e) Determine \(x\) such that the probability that you wait less than \(x\) minutes is \(0.50 .\)

The number of stork sightings on a route in South Carolina follows a Poisson process with a mean of 2.3 per year. (a) What is the mean time between sightings? (b) What is the probability that there are no sightings within three months \((0.25\) years \() ?\) (c) What is the probability that the time until the first sighting exceeds six months? (d) What is the probability of no sighting within three years?

The distance between major cracks in a highway follows an exponential distribution with a mean of five miles. (a) What is the probability that there are no major cracks in a 10 -mile stretch of the highway? (b) What is the probability that there are two major cracks in a 10 -mile stretch of the highway? (c) What is the standard deviation of the distance between major cracks? (d) What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection? (e) What is the probability that there are no major cracks in two separate five-mile stretches of the highway? (f) Given that there are no cracks in the first five miles inspected, what is the probability that there are no major cracks in the next 10 miles inspected?

The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours. (a) What is the probability that an assembly on test fails in less than 100 hours? (b) What is the probability that an assembly operates for more than 500 hours before failure? (c) If an assembly has been on test for 400 hours without a failure, what is the probability of a failure in the next 100 hours? (d) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently. (e) If 10 assemblies are tested, what is the probability that all have failed by 800 hours? Assume that the assemblies fail independently.

The time between arrivals of small aircraft at a county airport is exponentially distributed with a mean of one hour. (a) What is the probability that more than three aircraft arrive within an hour? (b) If 30 separate one-hour intervals are chosen, what is the probability that no interval contains more than three arrivals? (c) Determine the length of an interval of time (in hours) such that the probability that no arrivals occur during the interval is \(0.10 .\)

The time between calls to a corporate office is exponentially distributed with a mean of 10 minutes. (a) What is the probability that there are more than three calls in one-half hour? (b) What is the probability that there are no calls within onehalf hour? (c) Determine \(x\) such that the probability that there are no calls within \(x\) hours is 0.01 . (d) What is the probability that there are no calls within a twohour interval? (e) If four nonoverlapping one-half-hour intervals are selected, what is the probability that none of these intervals contains any call? (f) Explain the relationship between the results in part (a) and (b).

Assume that the flaws along a magnetic tape follow a Poisson distribution with a mean of 0.2 flaw per meter. Let \(X\) denote the distance between two successive flaws. (a) What is the mean of \(X ?\) (b) What is the probability that there are no flaws in \(10 \mathrm{con}-\) secutive meters of tape? (c) Does your answer to part (b) change if the 10 meters are not consecutive? (d) How many meters of tape need to be inspected so that the probability that at least one flaw is found is \(90 \% ?\) (e) What is the probability that the first time the distance between two flaws exceeds eight meters is at the fifth flaw? (f) What is the mean number of flaws before a distance between two flaws exceeds eight meters?

If the random variable \(X\) has an exponential distribution with mean \(\theta,\) determine the following: (a) \(P(X>\theta)\) (b) \(P(X>2 \theta)\) (c) \(P(X>3 \theta)\) (d) How do the results depend on \(\theta\) ?

Derive the formula for the mean and variance of an exponential random variable.

Web crawlers need to estimate the frequency of changes to Web sites to maintain a current index for Web searches. Assume that the changes to a Web site follow a Poisson process with a mean of 3.5 days. (a) What is the probability that the next change occurs in less than 2.0 days? (b) What is the probability that the time until the next change is greater 7.0 days? (c) What is the time of the next change that is exceeded with probability \(90 \%\) ? (d) What is the probability that the next change occurs in less than 10.0 days, given that it has not yet occurred after 3.0 days?

The length of stay at a specific emergency department in a hospital in Phoenix, Arizona had a mean of 4.6 hours. Assume that the length of stay is exponentially distributed. (a) What is the standard deviation of the length of stay? (b) What is the probability of a length of stay of more than 10 hours? (c) What length of stay is exceeded by \(25 \%\) of the visits?

An article in Journal of National Cancer Institute ["Breast Cancer Screening Policies in Developing Countries: A Cost-Effectiveness Analysis for India" \((2008,\) Vol. 100(18) pp. \(1290-1300\) ) ] presented a screening analysis model of breast cancer based on data from India. In this analysis, the time that a breast cancer case stays in a preclinical state is modeled to be exponentially distributed with a mean depending on the state. For example, the time that a cancer case stays in the state of \(\mathrm{T} 1 \mathrm{C}\) (tumor size of \(11-20 \mathrm{~mm}\) ) is exponentially distributed with a mean of 1.48 years. (a) What is the probability that a breast cancer case in India stays in the state of \(\mathrm{T} 1 \mathrm{C}\) for more than 2.0 years? (b) What is the proportion of breast cancer cases in India that spend at least 1.0 year in the state of \(\mathrm{T} 1 \mathrm{C} ?\) (c) Assume that a person in India is diagnosed to be in the state of \(\mathrm{T} 1 \mathrm{C}\). What is the probability that the patient is in the same state six months later?

An article in Vaccine ["Modeling the Effects of Influenza Vaccination of Health Care Workers in Hospital Departments" (2009, Vol.27(44), pp. \(6261-6267\) ) ] considered the immunization of healthcare workers to reduce the hazard rate of influenza virus infection for patients in regular hospital departments. In this analysis, each patient's length of stay in the department is taken as exponentially distributed with a mean of 7.0 days. (a) What is the probability that a patient stays in hospital for less than 5.5 days? (b) What is the probability that a patient stays in hospital for more than 10.0 days if the patient has currently stayed for 7.0 days? (c) Determine the mean length of stay such that the probability is 0.9 that a patient stays in the hospital less than 6.0 days.

An article in Ad Hoc Networks ["Underwater Acoustic Sensor Networks: Target Size Detection and Performance Analysis" \((2009,\) Vol. \(7(4),\) pp. \(803-808)]\) discussed an underwater acoustic sensor network to monitor a given area in an ocean. The network does not use cables and does not interfere with shipping activities. The arrival of clusters of signals generated by the same pulse is taken as a Poisson arrival process with a mean of \(\lambda\) per unit time. Suppose that for a specific underwater acoustic sensor network, this Poisson process has a rate of 2.5 arrivals per unit time. (a) What is the mean time between 2.0 consecutive arrivals? (b) What is the probability that there are no arrivals within 0.3 time units? (c) What is the probability that the time until the first arrival exceeds 1.0 unit of time? (d) Determine the mean arrival rate such that the probability is 0.9 that there are no arrivals in 0.3 time units.

Use the properties of the gamma function to evaluate the following: (a) \(\Gamma(6)\) (b) \(\Gamma(5 / 2)\) (c) \(\Gamma(9 / 2)\)