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introduction to probability statistics

Probability And Statistics For Engineering And The Sciences 8th Edition Jay L Devore, Roger Ellsbury - Solutions

Use the definition in Expression (3.13) to prove that. [Hint: With , where .]
Suppose and . What isa. E(X2)? [Hint:]?b. V(X)?c. The general relationship among the quantities E(X),, and V(X)?
Write a general rule for where c is a constant.What happens when you let , the expected value of X?
A result called Chebyshev’s inequality states that for any probability distribution of an rv X and any number k that is at least 1, . In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/k2.a. What is the value of the upper bound for ? ?? ?
If , show that .
Compute the following binomial probabilities directly from the formula for b(x; n, p):a. b(3; 8, .35)b. b(5; 8, .6)c. when andd. when and
Use Appendix Table A.1 to obtain the following probabilities:a. B(4; 15, .3)b. b(4; 15, .3)c. b(6; 15, .7)d. whene. whenf. when g. when
When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let of defective boards in a random sample of size , so .a. Determine .b. Determine .c. Determine .d. What is the probability that none of the 25 boards is defective?e.
Refer to the previous exercise.a. What is the expected number of calls among the 25 that involve a fax message?b. What is the standard deviation of the number among the 25 calls that involve a fax message?c. What is the probability that the number of calls among the 25 that involve a fax
Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider ran- domly selecting 25 purchasers.a. What are the mean value and standard deviation of the number who want a new copy of the book?b.
Exercise 30 (Section 3.3) gave the pmf of Y, the number of traffic citations for a randomly selected individual insured by a particular company. What is the probability that among 15 randomly chosen such individualsa. At least 10 have no citations?b. Fewer than half have at least one citation?c.
The College Board reports that 2% of the 2 million high school students who take the SAT each year receive special accommodations because of documented disabilities (Los Angeles Times, July 16, 2002). Consider a random sample of 25 students who have recently taken the test.a. What is the
Suppose that 90% of all batteries from a certain supplier have acceptable voltages. A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. Among ten randomly selected flashlights, what is the probability that at
A very large batch of components has arrived at a distribu- tor. The batch can be characterized as acceptable only if the proportion of defective components is at most .10. The distributor decides to randomly select 10 components and to accept the batch only if the number of defective components in
a. What is the probability that the batch will be accepted when the actual proportion of defectives is .01? .05?.10? .20?.25?b. Let p denote the actual proportion of defectives in the batch. A graph of P(batch is accepted) as a function of p. with p on the horizontal axis and P(batch is accepted)
c. Repeat parts (a) and (b) with "1" replacing "2" in the acceptance sampling plan.d. Repeat parts (a) and (b) with "15" replacing "10" in the acceptance sampling plan.e. Which of the three sampling plans, that of part (a), (c), or (d), appears most satisfactory, and why?
An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a par- ticular city for 1 year. The fire department is concerned that many houses remain without detectors. Let p = the true proportion of such houses having detectors, and suppose
a. What is the probability that the claim is rejected when the actual value of p is .8?b. What is the probability of not rejecting the claim when p = .7? When p = .6?c. How do the "error probabilities" of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14?
A toll bridge charges $1.00 for passenger cars and $2.50 for other vehicles. Suppose that during daytime hours, 60% of all vehicles are passenger cars. If 25 vehicles cross the bridge during a particular daytime period, what is the resulting expected toll revenue? [Hint: Let X = the number of
a. For fixed n, are there values of p (0 p 1) for which V(X) = 0? Explain why this is so.b. For what value of p is V(X) maximized? [Hint: Either graph (X) as a function of p or else take a derivative.]
a. Show that b(x; n, 1 - p) = b(n-x;n.p).b. Show that B(x; n, 1 - p) 1-B(n-x-1;n,p). [Hint: At most x S's is equivalent to at least (n - x) F's.]c. What do parts (a) and (b) imply about the necessity of including values of p greater than 5 in Appendix Table A.1?
Show that E(X) = np when X is a binomial random variable. [Hint: First express E(X) as a sum with lower limit x=1. Then factor out np. let y=x-1 so that the sum is from y = 0 to y=n-1, and show that the sum equals 1.]
Customers at a gas station pay with a credit card (4), debit card (B), or cash (C). Assume that successive customers make independent choices, with P(A) = .5, P(B) = .2, and P(C) =
a. Among the next 100 customers, what are the mean and variance of the number who pay with a debit card? Explain your reasoning.b. Answer part (a) for the number among the 100 who don't pay with cash.
An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reserva- tion. From previous records, 20% of all those making reservations do not appear for the trip. Answer the following
Refer to Chebyshev's inequality given in Exercise
Calculate P(X |ko) for k = 2 and k = 3 when X Bin(20, .5), and compare to the corresponding upper bound. Repeat for X ~ Bin(20, .75).
Each of 12 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscil- lating noise when the refrigerators are running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the
Twenty pairs of individuals playing in a bridge tournament have been seeded 1,...,
In the first part of the tourna- ment, the 20 are randomly divided into 10 east-west pairs and 10 north-south pairs.a. What is the probability that x of the top 10 pairs end up playing east-west?b. What is the probability that all of the top five pairs end up playing the same direction?c. If there
Suppose that p = P(male birth) = .5. A couple wishes to have exactly two female children in their family. They will have children until this condition is fulfilled.a. What is the probability that the family has x male children?b. What is the probability that the family has four children?c. What is
Let X, the number of flaws on the surface of a randomly selected boiler of a certain type, have a Poisson distribution with parameter =
Use Appendix Table A.2 to compute the following probabilities:a. P(X 8)d. P(5 X 8)b. P(X = 8)c. P(9 X)e. P(5 < x < 8)
Let X be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article "Methodology for Probabilistic Life Prediction of Multiple- Anomaly Materials" (Amer. Inst. of Aeronautics and Astronautics J., 2006: 787-793) proposes a Poisson distri- bution
a. Compute both P(X 4) and P(X < 4).b. Compute P(4 X 8).c. Compute P(81).d. What is the probability that the number of anomalies exceeds its mean value by no more than one standard deviation? = 20
Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter (suggested in the article "Dynamic Ride Sharing: Theory and Practice." J. of Transp. Engr., 1997: 308-312). What is the probability
Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses. Suppose this number X has a Poisson distribution with parameter
(Suggested in "Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribu- tion," J. Quality Technology, 1983: 126-129.)a. What is the probability that a disk has exactly one miss- ing pulse?b. What is the probability that a disk has at least two miss- ing pulses?c. If two disks
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate a 8 per hour, so that the number of arrivals during a time period of hours is a Poisson rv with parameter =
a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10?b. What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period?c. What is the probability that at least 20 small aircraft arrive
The number of people arriving for treatment at an emer- gency room can be modeled by a Poisson process with a rate parameter of five per hour.a. What is the probability that exactly four arrivals occur during a particular hour?b. What is the probability that at least four people arrive during a
Let X have a Poisson distribution with parameter . Show that E(X) = directly from the definition of expected value. [Hint: The first term in the sum equals 0, and then x can be can- celed. Now factor out and show that what is left sums to 1.]
Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parametera, the expected number of trees per acre, equal to
a. What is the probability that in a certain quarter-acre plot, there will be at most 16 trees?b. If the forest covers 85,000 acres, what is the expected number of trees in the forest?c. Suppose you select a point in the forest and construct a circle of radius .1 mile. Let X = the number of trees
Automobiles arrive at a vehicle equipment inspection sta- tion according to a Poisson process with rate a = 10 per hour. Suppose that with probability 5 an arriving vehicle will have no equipment violations.a. What is the probability that exactly ten arrive during the hour and all ten have no
a. In a Poisson process, what has to happen in both the time interval (0, 1) and the interval (t,+ Ar) so that no events occur in the entire interval (0, + Ar)? Use this and Assumptions 1-3 to write a relationship between Po(t + Ar) and P(t).b. Use the result of part (a) to write an expression for
c. Verify that P,(t) = et satisfies the equation of part (b).d. It can be shown in a manner similar to parts (a) and (b) that the P()s must satisfy the system of differential equations P(t) = aP (1) aP(1) k = 1,2,3,... Verify that P(t)=e(at)k! satisfies the system. (This is actually the only
Consider a deck consisting of seven cards, marked 1, 2,....
Three of these cards are selected at random. Define an rv W by W = the sum of the resulting numbers, and compute the pmf of W. Then compute and o. [Hint: Consider out- comes as unordered, so that (1, 3, 7) and (3, 1, 7) are not different outcomes. Then there are 35 outcomes, and they can be listed.
After shuffling a deck of 52 cards, a dealer deals out
Let X = the number of suits represented in the five-card hand.a. Show that the pmf of X is I 1 2 3 4 p(x) .002 .146 .588 .264 [Hint: p(1) 4P(all are spades), p(2) = 6P(only spades and hearts with at least one of each suit), and p(4) =4P(2 spades one of each other suit).]b. Compute , , and or.
A plan for an executive travelers' club has been developed by an airline on the premise that 10% of its current cus- tomers would qualify for membership.a. Assuming the validity of this premise, among 25 ran- domly selected current customers, what is the probabil- ity that between 2 and 6
What is the probability that the company's premise is rejected when it is actually valid?d. Refer to the decision rule introduced in part (c). What is the probability that the company's premise is not rejected even though p = .20 (i.e., 20% qualify)?
There are two Certified Public Accountants in a particular office who prepare tax returns for clients. Suppose that for a particular type of complex form, the number of errors made by the first preparer has a Poisson distribution with mean value, the number of errors made by the second preparer has
Use the fact that (x)p(x) = (x - p)p(x) all.x to prove Chebyshev's inequality given in Exercise
120. The simple Poisson process of Section 3.6 is characterized by a constant rate a at which events occur per unit time. A generalization of this is to suppose that the probability of exactly one event occurring in the interval [1,1 + Ar] is a(t) Ato(At). It can then be shown that the number of
Consider a collection 4,..., 4, of mutually exclusive and exhaustive events, and a random variable X whose distri- bution depends on which of the 4,'s occurs (e.g., a com- muter might select one of three possible routes from home to work, with X representing the commute time). Let E(X4) denote the
The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following density function: (x) = {.075.x (.075x + 2 3x5 otherwisea. Graph the pdf and verify that the total area under the den- sity curve is indeed
b. Calculate P(X4). How does this probability compare to P(X < 4)?c. Calculate P(3.5 X 4.5) and also P(4.5 < X).
Suppose the reaction temperature X (in C) in a certain chemical process has a uniform distribution with 4 = -5 and B =
a. Compute P(X < 0).b. Compute P(-2.5c. What is the probability that the lecture continues beyond the hour for between 60 and 90 sec?d. What is the probability that the lecture continues for at least 90 sec beyond the end of the hour?
The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a con- tinuous rv X with pdf f(x) (k[1-(x-3)] 2x4 0 otherwisea. Sketch the graph of f(x).b. Find the value of k.c. What is the probability that the actual tracking weight is
The time X (min) for a lab assistant to prepare the equipment for a certain experiment is believed to have a uniform distri- bution with A=25 and B =
a. Determine the pdf of X and sketch the corresponding density curve.b. What is the probability that preparation time exceeds 33 min?c. What is the probability that preparation time is within 2 min of the mean time? [Hint: Identify from the graph of f(x).]d. For any a such that 25 < a
In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the wait- ing time (in minutes) at each stop has a uniform distribution with 4 0 and B =5, then it can be shown that the total waiting time Y has the pdfa. Sketch a graph of the pdf of
c. What is the probability that total waiting time is at most 3 min?d. What is the probability that total waiting time is at most 8 min?e. What is the probability that total waiting time is between 3 and 8 min?f. What is the probability that total waiting time is either less than 2 min or more than
Consider again the pdf of X = time headway given in Example 4.5. What is the probability that time headway isa. At most 6 sec?b. More than 6 sec? At least 6 sec?c. Between 5 and 6 sec?
A family of pdf's that has been used to approximate the dis- tribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, k and
both 0, and the pdf is f(x;k, 0)=x+1a. Sketch the graph of f(x;k, 0). *NG 0 x < 0b. Verify that the total area under the graph equals
c. If the rv X has pdf f(x; k, 0), for any fixed b > 0, obtain an expression for P(X = b).d. For
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is (0x
The cdf for X (= measurement error) of Exercise 3 is 0 x 1.5)d. The median checkout duration [solve .5 = F()]e. F'(x) to obtain the density function f(x)f. E(X) g. V(X) and oy h. If the borrower is charged an amount h(x) = x when checkout duration is X, compute the expected charge E[h(X)].a.
Let X denote the amount of space occupied by an article placed in a 1-ft packing container. The pdf of X is 90x (1-x) 0a. Graph the pdf. Then obtain the cdf of X and graph it.b. What is P(X5) [i.e., F(.5)]?c. Using the cdf from (a), what is P(.25 < x < .5)? What is P(.25 X .5)?d. What is the 75th
Answer parts (a)-(f) of Exercise 15 with X = lecture time past the hour given in Exercise
17. Let X have a uniform distribution on the interval [A, B].a. Obtain an expression for the (100p)th percentile.b. Compute E(X), V(X), andc. For n, a positive integer, compute E(X").
Let X denote the voltage at the output of a microphone, and suppose that X has a uniform distribution on the interval from -1 to
The voltage is processed by a "hard limiter" with cutoff values-5 and .5, so the limiter output is a ran- dom variable Y related to X by Y = Xif | .5, Y = .5 if X>5, and Y=-5 if X < -5.a. What is P(Y .5)?b. Obtain the cumulative distribution function of Y and graph it.
Let X be a continuous rv with cdf 0 x 0 F(x) + In 0 < x 4 x>4 [This type of cdf is suggested in the article "Variability in Measured Bedload-Transport Rates" (Water Resources Bull., 1985: 39-48) as a model for a certain hydrologic vari- able.] What isa. P(X 1)?b. P(1 X 3)?c. The pdf of X?
Consider the pdf for total waiting time Y for two buses 25 0y
a. Compute and sketch the cdf of Y. [Hint: Consider sepa- rately 0 y
If the temperature at which a certain compound melts is a random variable with mean value 120C and standard devi- ation 2C, what are the mean temperature and standard deviation measured in F? [Hint: F = 1.8C + 32.]
Let X have the Pareto pdf k-8k f(x;k, 0) = 0 x
a. If k > 1, compute E(X).b. What can you say about E(X) if k = 1?c. If k >2, show that V(X) = ko(k-1)-2 (k-2)-1.d. If k = 2, what can you say about V(X)?e. What conditions on k are necessary to ensure that E(X") is finite?
Let X be the total medical expenses (in 1000s of dollars) incurred by a particular individual during a given year. Although X is a discrete random variable, suppose its distri- bution is quite well approximated by a continuous distribu- tion with pdf f(x) = k(1 + x/2.5)-7 for x
a. What is the value of K?b. Graph the pdf of X.c. What are the expected value and standard deviation of total medical expenses?d. This individual is covered by an insurance plan that entails a $500 deductible provision (so the first $500 worth of expenses are paid by the individual). Then the plan
When a dart is thrown at a circular target, consider the loca- tion of the landing point relative to the bull's eye. Let X be the angle in degrees measured from the horizontal, and assume that X is uniformly distributed on [0, 360]. Define Y to be the transformed variable Y = h(x) = (2/360)X - , so
and then
Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate.a. P(0 Z 2.17)c. P(-2.50Z 0)e. P(Z1.37) g. P(-1.50 Z 2.00) i. P(1.50 Z)b. P(0 Z 1)d. P(-2.50 Z 2.50)f. P(-1.75 Z) h. P(1.37 Z 2.50) j. P(|Z 2.50)
In each case, determine the value of the constant c that makes the probability statement correct.a. (c) = .9838c. Plc Z) .121e. P(c|Z) = .016b. P(0 Zc) = .291d. P(-c Zc) = .668
Determine for the following:a. a = .0055b. a=.09c. a = .663
Suppose the force acting on a column that helps to support a building is a normally distributed random variable X with mean value 15.0 kips and standard deviation 1.25 kips. Compute the following probabilities by standardizing and then using Table A.3.a. P(X 15)c. P(X 10)e. P(X - 15 3)b. P(X
Spray drift is a constant concern for pesticide applicators and agricultural producers. The inverse relationship between droplet size and drift potential is well known. The paper "Effects of 2,4-D Formulation and Quinclorac on Spray Droplet Size and Deposition" (Weed Technology, 2005: 1030-1036)
a. If a normal distribution has 30 and or = 5, what is the 91st percentile of the distribution?b. What is the 6th percentile of the distribution?c. The width of a line etched on an integrated circuit chip is normally distributed with mean 3.000 m and standard deviation .140. What width value
The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.256 ohms and 5% having a resistance smaller than 9.671 ohms. What are the mean value and standard deviation of the resistance dis- tribution?
A machine that produces ball bearings has initially been set so that the true average diameter of the bearings it pro- duces is .500 in. A bearing is acceptable if its diameter is within .004 in. of this target value. Suppose, however, that the setting has changed during the course of production,
The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 70 and standard deviation
(Rockwell hardness is measured on a contin- uous scale.)a. If a specimen is acceptable only if its hardness is between 67 and 75, what is the probability that a ran- domly chosen specimen has an acceptable hardness?b. If the acceptable range of hardness is (70 -c, 70 + c), for what value of c would
Suppose Appendix Table A.3 contained (2) only for
Explain how you could still computea. P(-1.72 Z -.55)b. P(-1.72 Z .55) Is it necessary to tabulate (2) for negative? What prop- erty of the standard normal curve justifies your answer?
Chebyshev's inequality, (see Exercise 44, Chapter 3), is valid for continuous as well as discrete distributions. It states that for any number k satisfying k 1, P(X-kor) 1/ (see Exercise 44 in Chapter 3 for an interpretation). Obtain this probability in the case of a normal distribution for k = 1,

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