Page 173 - Applied statistics and probability for engineers

Section 4-12/Beta Distribution 151

4-190. An allele is an alternate form of a gene, and the propor- uncertainties in inancial resources. The completion time for the
tion of alleles in a population is of interest in genetics. An article irst phase is modeled with a beta distribution and the minimum,
in BMC Genetics [“Calculating Expected DNA Remnants From most likely (mode), and maximum completion times for the irst
Ancient Founding Events in Human Population Genetics” (2008, phase are 1.0, 1.25, and 2.0 years, respectively. Also, the mean
1
6
Vol. 9:66)] used a beta distribution with mean 0.3 and standard time is assumed to equal μ = +1 4 1( .25) + 2) / = .333. Deter-
deviation 0.17 to model initial allele proportions in a genetic simu- mine the following in parts (a) and (b):
lation. Determine the parameters α and β for this beta distribution. (a) Parameters α and β of the beta distribution.
4-191. Suppose that the construction of a solar power station is (b) Standard deviation of the distribution.
initiated. The project’s completion time has not been set due to (c) Sketch the probability density function.

Supplemental Exercises

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.

4-192. The probability density function of the time it (e) What is the expected number of the 10 cases that are
takes a hematology cell counter to complete a test on a blood between 89.7 and 90.3 millimeters?
sample is f x ( ) = .04 for 50 < x < 75 seconds. 4-196. The sick-leave time of employees in a irm in a
0
(a) What percentage of tests requires more than 70 seconds to month is normally distributed with a mean of 100 hours and a
complete? standard deviation of 20 hours.
(b) What percentage of tests requires less than one minute to (a) What is the probability that the sick-leave time for next
complete? month will be between 50 and 80 hours?
(c) Determine the mean and variance of the time to complete a (b) How much time should be budgeted for sick leave if the
test on a sample. budgeted amount should be exceeded with a probability of
4-193. The tensile strength of paper is modeled by a nor- only 10%?
mal distribution with a mean of 35 pounds per square inch and 4-197. The percentage of people exposed to a bacteria who
a standard deviation of 2 pounds per square inch. become ill is 20%. Assume that people are independent. Assume
that 1000 people are exposed to the bacteria. Approximate each of
(a) What is the probability that the strength of a sample is less
2
than 40 lb/in ? the following:
(a) Probability that more than 225 become ill
(b) If the speciications require the tensile strength to exceed
2
30 lb/in , what proportion of the samples is scrapped? (b) Probability that between 175 and 225 become ill
(c) Value such that the probability that the number of people
4-194. The time it takes a cell to divide (called mitosis) who become ill exceeds the value is 0.01
is normally distributed with an average time of one hour and a 4-198. The time to failure (in hours) for a laser in a cytom-
standard deviation of ive minutes. etry machine is modeled by an exponential distribution with
(a) What is the probability that a cell divides in less than 45 λ = .00004 . What is the probability that the time until failure is
0
minutes? (a) At least 20,000 hours? (b) At most 30,000 hours?
(b) What is the probability that it takes a cell more than 65 (c) Between 20,000 and 30,000 hours?
minutes to divide? 4-199. When a bus service reduces fares, a particular trip
(c) By what time have approximately 99% of all cells com- from New York City to Albany, New York, is very popular. A
pleted mitosis? small bus can carry four passengers. The time between calls for
4-195. The length of an injection-molded plastic case tickets is exponentially distributed with a mean of 30 minutes.
that holds magnetic tape is normally distributed with a length Assume that each caller orders one ticket. What is the probabil-
of 90.2 millimeters and a standard deviation of 0.1 millimeter. ity that the bus is illed in less than three hours from the time
(a) What is the probability that a part is longer than 90.3 mil- of the fare reduction?
limeters or shorter than 89.7 millimeters? 4-200. The time between process problems in a manufac-
(b) What should the process mean be set at to obtain the high- turing line is exponentially distributed with a mean of 30 days.
est number of parts between 89.7 and 90.3 millimeters? (a) What is the expected time until the fourth problem?
(c) If parts that are not between 89.7 and 90.3 millimeters are (b) What is the probability that the time until the fourth prob-
scrapped, what is the yield for the process mean that you lem exceeds 120 days?
selected in part (b)? 4-201. The life of a recirculating pump follows a Weibull dis-
Assume that the process is centered so that the mean is 90 milli- tribution with parameters β = 2 and δ = 700 hours. Determine
meters and the standard deviation is 0.1 millimeter. Suppose that for parts (a) and (b):
10 cases are measured, and they are assumed to be independent. (a) Mean life of a pump (b) Variance of the life of a pump
(d) What is the probability that all 10 cases are between 89.7 (c) What is the probability that a pump will last longer than its
and 90.3 millimeters? mean?