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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

Buses arrive at a specified stop at 15-minute intervals starting at 7 A.M. That is, they arrive at 7, 7:15, 7:30, 7:45, and so on. If a passenger arrives at the stop at a time that is uniformly
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
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
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
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
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
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
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
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
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
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
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
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
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
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 −
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
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)
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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.) ==
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
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
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)
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)
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
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
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 )
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)
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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,
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
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.
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
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
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
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.
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)
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,
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.

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