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introduction to probability statistics

Introduction To Probability And Statistics For Engineers And Scientists 3rd Edition Sheldon M. Ross - Solutions

The current in a semiconductor diode is often measured by the Shockley equationwhere V is the voltage across the diode; I0 is the reverse current; a is a constant; and I is the resulting diode current. Find E[I] if a = 5, I0 = 10−6, and V is uniformly distributed over (1, 3). I = Io (ev - 1)
If X is a normal random variable with mean μ = 3 and varianceσ2 = 16, find(a) P{X < 11};(b) P{X > −1};(c) P{2 < X < 7}.
The power W dissipated in a resistor is proportional to the square of the voltage V. That is, W = rV 2where r is a constant. If r = 3, and V can be assumed (to a very good approximation) to be a normal random variable with mean 6 and standard deviation 1, find (a) E [W];(b) P{W > 120}.
Data from the National Oceanic and Atmospheric Administration indicate that the yearly precipitation in Los Angeles is a normal random variable with a mean of 12.08 inches and a standard deviation of 3.1 inches.(a) Find the probability that the total precipitation during the next 2 years will
Suppose that a number of miles that a car can run before its battery wears out is exponentially distributed with an average value of 10,000 miles. If a person desires to take a 5,000-mile trip, what is the probability that she will be able to complete her trip without having to replace her car
A crew of workers has 3 interchangeable machines, of which 2 must be working for the crew to do its job. When in use, each machine will function for an exponentially distributed time having parameter λ before breaking down. The workers decide to initially use machines A and B and keep machine C in
A series system is one that needs all of its components to function in order for the system itself to be functional. For an n-component series system in which the component lifetimes are independent exponential random variables with respective parameters λ1, λ2, . . . , λn, what is the
EXAMPLE 5.7a The lifetime of a battery is exponentially distributed with rate λ. If a stereo cassette requires one battery to operate, then the total playing time one can obtain from a total of n batteries is a gamma random variable with parameters (n, λ). ■Figure 5.11 presents a graph of the
Determine is a chi-square random variable with 26 degrees of freedom. P{X630) when x26
Find 2 X.05.15
Suppose that we are attempting to locate a target in three-dimensional space, and that the three coordinate errors (in meters) of the point chosen are independent normal random variables with mean 0 and standard deviation 2. Find the probability that the distance between the point chosen and the
When we attempt to locate a target in two-dimensional space, suppose that the coordinate errors are independent normal random variables with mean 0 and standard deviation 2. Find the probability that the distance between the point chosen and the target exceeds 3.
Find (a) P{T12 ≤ 1.4} and (b) t.025,9.
Determine P{F6,14 1.5).
1. A satellite system consists of 4 components and can function adequately if at least 2 of the 4 components are in working condition. If each component is, independently, in working condition with probability .6, what is the probability that the system functions adequately?
2. A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability .2. Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit 00000 instead of 0
3. If each voter is for Proposition A with probability .7, what is the probability that exactly 7 of 10 voters are for this proposition?
4. Suppose that a particular trait (such as eye color or left-handedness) of a person is classified on the basis of one pair of genes, and suppose that d represents a dominant gene and r a recessive gene. Thus, a person with dd genes is pure dominance, one with rr is pure recessive, and one with rd
5. At least one-half of an airplane’s engines are required to function in order for it to operate. If each engine independently functions with probability p, for what values of p is a 4-engine plane more likely to operate than a 2-engine plane?
6. Let X be a binomial random variable withFind (a) P{X = 4};(b) P{X > 12}. E[X] 7 and Var(X) = 2.1.
7. If X and Y are binomial random variables with respective parameters (n, p) and(n, 1 − p), verify and explain the following identities:(a) P{X ≤ i} = P{Y ≥ n − i};(a) P{X = k} = P{Y = n − k}.
8. If X is a binomial random variable with parameters n and p, where 0 (b) As k goes from 0 to n, P{X = k} first increases and then decreases, reaching its largest value when k is the largest integer less than or equal to (n + 1)p. (a) P{X=k+1} = P n-k 1-pk+1 P(X=k}, k = 0, 1,...,n-1.
9. Derive the moment generating function of a binomial random variable and then use your result to verify the formulas for the mean and variance given in the text.
10. Compare the Poisson approximation with the correct binomial probability for the following cases:(a) P{X = 2} when n = 10, p = .1;(b) P{X = 0} when n = 10, p = .1;(c) P{X = 4} when n = 9, p = .2.
11. If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1 100 , what is the (approximate) probability that you will win a prize (a)at least once, (b) exactly once, and (c) at least twice?
12. The number of times that an individual contracts a cold in a given year is a Poisson random variable with parameter λ = 3. Suppose a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to λ = 2 for 75 percent of the population.
13. In the 1980s, an average of 121.95 workers died on the job each week. Give estimates of the following quantities:(a) the proportion of weeks having 130 deaths or more;(b) the proportion of weeks having 100 deaths or less.Explain your reasoning.
15. The game of frustration solitaire is played by turning the cards of a randomly shuffled deck of 52 playing cards over one at a time. Before you turn over the first card, say ace; before you turn over the second card, say two, before you turn over the third card, say three. Continue in this
16. The probability of error in the transmission of a binary digit over a communication channel is 1/103. Write an expression for the exact probability of more than 3 errors when transmitting a block of 103 bits. What is its approximate value?Assume independence.
17. If X is a Poisson random variable with mean λ, show that P{X = i } first increases and then decreases as i increases, reaching its maximum value when i is the largest integer less than or equal to λ.
18. A contractor purchases a shipment of 100 transistors. It is his policy to test 10 of these transistors and to keep the shipment only if at least 9 of the 10 are in working condition. If the shipment contains 20 defective transistors, what is the probability it will be kept?
19. Let X denote a hypergeometric random variable with parameters n, m, and k.That is, (a) Derive a formula for P{X = i} in terms of P{X = i − 1}.(b) Use part (a) to compute P{X = i} for i = 0, 1, 2, 3, 4, 5 when n = m = 10, k = 5, by starting with P{X = 0}.(c) Based on the recursion in part
20. Independent trials, each of which is a success with probability p, are successively performed. Let X denote the first trial resulting in a success. That is, X will equal k if the first k −1 trials are all failures and the kth a success. X is called a geometric random variable. Compute(a) P{X
21. If U is uniformly distributed on (0, 1), show that a + (b − a)U is uniform on (a, b).
22. You arrive at a bus stop at 10 o’clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. What is the probability that you will have to wait longer than 10 minutes? If at 10:15 the bus has not yet arrived, what is the probability that you will have to
23. If X is a normal random variable with parameters μ = 10, σ2 = 36, compute(a) P{X > 5};(b) P{4 < X < 16};(c) P{X < 8};(d) P{X < 20};(e) P{X > 16}.
24. The Scholastic Aptitude Test mathematics test scores across the population of high school seniors follow a normal distribution with mean 500 and standard deviation 100. If five seniors are randomly chosen, find the probability that(a) all scored below 600 and (b) exactly three of them scored
25. The annual rainfall (in inches) in a certain region is normally distributed withμ = 40, σ = 4. What is the probability that in 2 of the next 4 years the rainfall will exceed 50 inches? Assume that the rainfalls in different years are independent.
26. The width of a slot of a duralumin forging is (in inches) normally distributed withμ = .9000 and σ = .0030. The specification limits were given as .9000±.0050.What percentage of forgings will be defective? What is the maximum allowable value of σ that will permit no more than 1 in 100
27. A certain type of lightbulb has an output that is normally distributed with mean 2,000 end foot candles and standard deviation 85 end foot candles. Determine a lower specification limit L so that only 5 percent of the lightbulbs produced will be defective. (That is, determine L so that P{X ≥
28. A manufacturer produces bolts that are specified to be between 1.19 and 1.21 inches in diameter. If its production process results in a bolt’s diameter being normally distributed with mean 1.20 inches and standard deviation .005, what percentage of bolts will not meet specifications?
29. Let (a) Show that for any μ and σand then evaluating the double integral by means of a change of variables to polar coordinates. (That is, let x = r cos θ, y = r sin θ, dx dy = r dr dθ.) == I=fe-x12 dx.
30. A random variable X is said to have a lognormal distribution if log X is normally distributed. If X is lognormal with E[log X] = μ and Var(log X ) = σ2, determine the distribution function of X. That is, what is P{X ≤ x}?
31. The lifetimes of interactive computer chips produced by a certain semiconductor manufacturer are normally distributed having mean 4.4 × 106 hours with a standard deviation of 3 × 105 hours. If a mainframe manufacturer requires that at least 90 percent of the chips from a large batch will have
32. In Problem 31, what is the probability that a batch of 100 chips will contain at least 4 whose lifetimes are less than 3.8 × 106 hours?
33. The lifetime of a color television picture tube is a normal random variable with mean 8.2 years and standard deviation 1.4 years. What percentage of such tubes lasts(a) more than 10 years;(b) less than 5 years;(c) between 5 and 10 years?
34. The annual rainfall in Cincinnati is normally distributed with mean 40.14 inches and standard deviation 8.7 inches.(a) What is the probability this year’s rainfall will exceed 42 inches?(b) What is the probability that the sum of the next 2 years’ rainfall will exceed 84 inches?(c) What is
35. The height of adult women in the United States is normally distributed with mean 64.5 inches and standard deviation 2.4 inches. Find the probability that a randomly chosen woman is(a) less than 63 inches tall;(b) less than 70 inches tall;(c) between 63 and 70 inches tall.(d) Alice is 72 inches
36. An IQ test produces scores that are normally distributed with mean value 100 and standard deviation 14.2. The top 1 percent of all scores are in what range?
37. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ = 1.(a) What is the probability that a repair time exceeds 2 hours?(b) What is the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2
38. The number of years a radio functions is exponentially distributed with parameterλ = 18. If Jones buys a used radio, what is the probability that it will be working after an additional 10 years?
40. Let X1, X2, . . . , Xn denote the first n interarrival times of a Poisson process and set Sn = ni=1 Xi .(a) What is the interpretation of Sn?(b) Argue that the two events {Sn ≤ t } and {N(t ) ≥ n} are identical.(c) Use part (b) to show thatd) By differentiating the distribution function of
41. Earthquakes occur in a given region in accordance with a Poisson process with rate 5 per year.(a) What is the probability there will be at least two earthquakes in the first half of 2010?(b) Assuming that the event in part (a) occurs, what is the probability that there will be no earthquakes
42. When shooting at a target in a two-dimensional plane, suppose that the horizontal miss distance is normally distributed with mean 0 and variance 4 and is independent of the vertical miss distance, which is also normally distributed with mean 0 and variance 4. Let D denote the distance between
43. If X is a chi-square random variable with 6 degrees of freedom, find(a) P{X ≤ 6};(b) P{3 ≤ X ≤ 9}.
44. If X and Y are independent chi-square random variables with 3 and 6 degrees of freedom, respectively, determine the probability that X + Y will exceed 10.
45. Show that (1/2) =√π (Hint: Evaluate∞0 e−xx−1/2 dx by letting x = y2/2, dx = y dy.)
46. If T has a t-distribution with 8 degrees of freedom, find (a) P{T ≥ 1},(b) P{T ≤ 2}, and (c) P{−1 < T < 1}.
47. If Tn has a t -distribution with n degrees of freedom, show that T 2 n has an F -distribution with 1 and n degrees of freedom.
48. Let be the standard normal distribution function. If, for constants a and b > 0characterize the distribution of X. P{X x}=\ x-a b
Table 2.5 gives the monthly and yearly average daily minimum temperatures in 35 U.S. cities.The annual average daily minimum temperatures from Table 2.5 are represented in the following stem and leaf plot. 7 0.0 6 9.0 5 1.0, 1.3, 2.0, 5.5, 7.1, 7.4, 7.6, 8.5, 9.3 4 0.0, 1.0, 2.4, 3.6, 3.7, 4.8,
The winning scores in the U.S. Masters golf tournament in the years from 1982 to 1991 were as follows:284, 280, 277, 282, 279, 285, 281, 283, 278, 277 Find the sample mean of these scores.
The following is a frequency table giving the ages of members of a symphony orchestra for young adults.Find the sample mean of the ages of the 54 members of the symphony. Age Frequency 2 25 15 16 5 17 11 18 19 943 14 20 13
Find the sample median for the data described in Example 2.3b.
EXAMPLE 2.3d In a study reported in Hoel, D. G., “A representation of mortality data by competing risks,” Biometrics, 28, pp. 475–488, 1972, a group of 5-week-old mice were each given a radiation dose of 300 rad. The mice were then divided into two groups;the first group was kept in a
The following frequency table gives the values obtained in 40 rolls of a die.Find (a) the sample mean, (b) the sample median, and (c) the sample mode. Value Frequency - 1 9 5 6 85567 3 5 23456 7
Find the sample variances of the data sets A and B given below. A: 3,4,6,7, 10 B: -20,5, 15, 24
The following data give the worldwide number of fatal airline accidents of commercially scheduled air transports in the years from 1985 to 1993.Find the sample variance of the number of accidents in these years. Year 1985 1986 1987 1988 1989 1990 1991 1992 1993 22 22 26 28 27 25 30 29 24 Accidents
Table 2.6 lists the populations of the 25 most populous U.S. cities for the year 1994. For this data set, find (a) the sample 10 percentile and (b) the sample 80 percentile.
Noise is measured in decibels, denoted as dB. One decibel is about the level of the weakest sound that can be heard in a quiet surrounding by someone with good hearing; a whisper measures about 30 dB; a human voice in normal conversation is about 70 dB; a loud radio is about 100 dB. Ear discomfort
Table 2.7 lists the 10 top-selling passenger cars in the United States in 1999. A simple calculation gives that the sample mean and sample standard deviation ofthese data areThus Chebyshev’s inequality yields that at least 100(5/9)= 55.55 percent of the data lies in the interval TABLE 2.7 Top 10
The following stem and leaf plot gives the scores on a statistics exam taken by industrial engineering students.By standing the stem and leaf plot on its side we can see that the corresponding histogram is approximately normal. Use it to assess the empirical rule. 9 0, 1,4 8 3,5,5,7,8 7
Find the sample correlation coefficient for the data presented in Table 2.8.
The following data give the resting pulse rates (in beats per minute) and the years of schooling of 10 individuals. A scatter diagram of these data is presented in Figure 2.15. The sample correlation coefficient for these data is r =−.7638. This negative correlation indicates that for this data
1. The following is a sample of prices, rounded to the nearest cent, charged per gallon of standard unleaded gasoline in the San Francisco Bay area in June 1997.137, 139, 141, 137, 144, 141, 139, 137, 144, 141, 143, 143, 141 Represent these data in(a) a frequency table;(b) a relative frequency line
2. Explain how a pie chart can be constructed. If a data value had relative frequency r, at what angle would the lines defining its sector meet?
3. The following are the estimated oil reserves, in billions of barrels, for four regions in the western hemisphere.Represent these data in a pie chart. United States 38.7 South America 22.6 Canada 8.8 Mexico 60.0
4. The following table gives the average travel time to work for workers in each of the 50 states as well as the percentage of those workers who use public transportation.(a) Represent the data relating to the average travel times in a histogram.(b) Represent the data relating to the percentage of
5. Choose a book or article and count the number of words in each of the first 100 sentences. Present the data in a stem and leaf plot. Now choose another book or article, by a different author, and do the same. Do the two stem and leaf plots look similar? Do you think this could be a viable method
6. The following table gives the number of commercial airline accidents and fatalities in the United States in the years from 1980 to 1995.(a) Represent the number of yearly airline accidents in a frequency table.(b) Give a frequency polygon graph of the number of yearly airline accidents.(c) Give
7. (Use the table from Problem 6.)(a) Represent the number of yearly airline fatalities in a histogram.(b) Represent the number of yearly airline fatalities in a stem and leaf plot.(c) Find the sample mean of the number of yearly airline fatalities.(d) Find the sample median of the number of yearly
8. The following table gives the winning scores in the Masters golf tournament for the years from 1967 to 2002. Use it(a) to construct a stem and leaf plot, and(b) to find the sample median of the winning scores in these years. Year Player Score 1967 Gay Brewer, Jr. 280 1968 Bob Goalby 277 1969
9. Using the table given in Problem 4, find the sample mean and sample median of the average travel time for those states in the(a) northeast;(b) midwest;(c) south;(d) west.
10. The following data are the median prices for single-family homes in a variety of American cities for the years 1992 and 1994.(a) Represent the 1992 data in a histogram.(b) Represent the 1994 data in a stem and leaf plot.(c) Find the sample median of these median prices for 1992.(d) Find the
11. The following table gives the number of pedestrians, classified according to age group and sex, killed in fatal road accidents in England in 1922.(a) Approximate the sample means of the ages of the males.(b) Approximate the sample means of the ages of the females.(c) Approximate the quartiles
12. The following are the percentages of ash content in 12 samples of coal found in close proximity:9.2, 14.1, 9.8, 12.4, 16.0, 12.6, 22.7, 18.9, 21.0, 14.5, 20.4, 16.9 Find the(a) sample mean, and(b) sample standard deviation of these percentages.
13. Using the table given in Problem 4, find the sample variance of the average travel time for those states in the(a) south Atlantic;(b) mountain region.
14. The sample mean and sample variance of five data values are, respectively, ¯x =104 and s2 =4. If three of the data values are 102, 100, 105, what are the other two data values?
15. The following table gives the average annual pay, per state, in the years 1992 and 1993.(a) Do you think that the sample mean of the averages for the 50 states will equal the value given for the entire United States?(b) If the answer to part (a) is no, explain what other information aside from
16. The following data represent the lifetimes (in hours) of a sample of 40 transistors:(a) Determine the sample mean, median, and mode.(b) Give a cumulative relative frequency plot of these data. 112, 121, 126, 108, 141, 104, 136, 134 121, 118, 143, 116, 108, 122, 127, 140 113, 117, 126, 130, 134,
17. An experiment measuring the percent shrinkage on drying of 50 clay specimens produced the following data:(a) Draw a stem and leaf plot of these data.(b) Compute the sample mean, median, and mode.(c) Compute the sample variance.(d) Group the data into class intervals of size 1 percent starting
18. A computationally efficient way to compute the sample mean and sample variance of the data set x1, x2, . . . , xn is as follows. Letbe the sample mean of the first j data values; and letbe the sample variance of the first j, j ≥ 2, values. Then, with s2 1 =0, it can be shown that(a) Use the
19. Use the data concerning the prices of single-family homes provided in Problem 10 to find the(a) 10 percentile of the median prices;(b) 40 percentile of the median prices;(c) 90 percentile of the median prices.
20. Use the following table to find the quartiles of the average annual pay in the specified areas. Average Annual Pay by New York State Metropolitan Areas, 1999 Rank Amt. Rank Amt. Albany-Sch'dy-Troy $31,901 Nassau-Suffolk $36,944 Binghamton 29,167 New York City 52,351 Buffalo-Niagara Falls 30,487
21. Use the following figure, which gives the amounts of federal research money given to 15 universities in 1992, to answer this problem.(a) Which universities were given more than $225 million?(b) Approximate the sample mean of the amounts given to these universities.(c) Approximate the sample
22. Use the part of the table given in Problem 4 that gives the percentage of workers in each state that use public transportation to get to work to draw a box plot of these 50 percentages.
23. The following table gives the numbers of dogs, categorized by breed, registered in the American Kennel Club in 2000. Represent these numbers in a box plot.
24. The average particulate concentration, in micrograms per cubic meter, was measured in a petrochemical complex at 36 randomly chosen times, with the following concentrations resulting:5, 18, 15, 7, 23, 220, 130, 85, 103, 25, 80, 7, 24, 6, 13, 65, 37, 25, 24, 65, 82, 95, 77, 15, 70, 110, 44, 28,
25. A chemical engineer desiring to study the evaporation rate of water from brine evaporation beds obtained data on the number of inches of evaporation in each of 55 July days spread over 4 years. The data are given in the following stem and leaf plot, which shows that the smallest data value was
26. The following are the grade point averages of 30 students recently admitted to the graduate program in the Department of Industrial Engineering and Operations Research at the University of California at Berkeley.3.46, 3.72, 3.95, 3.55, 3.62, 3.80, 3.86, 3.71, 3.56, 3.49, 3.96, 3.90, 3.70, 3.61,

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