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introduction to probability statistics

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

Let X denote the number of flaws along a 100-m reel of magnetic tape (an integer-valued variable). Suppose X has approximately a normal distribution with = 25 and =
Use the continuity correction to calculate the prob- ability that the number of flaws isa. Between 20 and 30, inclusive.b. At most
Less than
There is no nice formula for the standard normal cdf (=), but several good approximations have been published in art- icles. The following is from "Approximations for Hand Calculators Using Small Integer Coefficients" (Mathematics of Computation, 1977: 214-222). For 0
The paper "Microwave Observations of Daily Antarctic Sea-Ice Edge Expansion and Contribution Rates" (IEEE Geosci. and Remote Sensing Letters, 2006: 54-58) states that "The distribution of the daily sea-ice advance/retreat from each sensor is similar and is approximately double exponential." The
Evaluate the following:a. I(6)b. I'(5/2)c. F(4; 5) (the incomplete gamma function)d. F(5:4)e. F(0; 4)
Let X have a standard gamma distribution with a =
Evaluate the following:a. P(X 5)d. P(3 X 8)b. P(X 8)e. P(3 6)
Suppose the time spent by a randomly selected student who uses a terminal connected to a local time-sharing computer facility has a gamma distribution with mean 20 min and variance 80 min.a. What are the values of a and ?b. What is the probability that a student uses the terminal for at most 24
The special case of the gamma distribution in which a is a positive integer n is called an Erlang distribution. If we replace by 1/A in Expression (4.8), the Erlang pdf is f(x; A, n)= A(Ax)-Ax (n-1)! 0 x
A system consists of five identical components connected in series as shown: 2 5 As soon as one component fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with A= .01 and that components fail inde- pendently of one another. Define events 4,
a. The event {y} is equivalent to what event involv- ing X itself?b. If X has a standard normal distribution, use part (a) to write the integral that equals P(X2y). Then differenti- ate this with respect to y to obtain the pdf of X [the square of a N(0, 1) variable]. Finally, show that X has a
The lifetime X (in hundreds of hours) of a certain type of vacuum tube has a Weibull distribution with parameters a = 2 and
Compute the following:a. E(X) and V(X)b. P(X 6)c. P(1.5 X 6) (This Weibull distribution is suggested as a model for time in service in "On the Assessment of Equipment Reliability: Trading Data Collection Costs for Precision," J. of Engr. Manuf., 1991: 105-109.)
Let X = the time (in 10-1 weeks) from shipment of a defec- tive product until the customer returns the product. Suppose that the minimum return time is y = 3.5 and that the excess X-3.5 over the minimum has a Weibull distribution with parameters a 2 and 1.5 (see "Practical Applications of the
Let X have a Weibull distribution with the pdf from Expression (4.11). Verify that BI(1 + 1/a). [Hint: In the integral for E(X). make the change of variable y=(x/B)", so that x = Byl]
a. In Exercise 72, what is the median lifetime of such tubes? [Hint: Use Expression (4.12).]b. In Exercise 74, what is the median return time?c. If X has a Weibull distribution with the cdf from Expression (4.12), obtain a general expression for the (100p)th percentile of the distribution.d. In
The authors of the paper from which the data in Exercise 1.27 was extracted suggested that a reasonable probability model for drill lifetime was a lognormal distribution with 4.5 and
a. What are the mean value and standard deviation of lifetime?b. What is the probability that lifetime is at most 100?c. What is the probability that lifetime is at least 200? Greater than 200?
A theoretical justification based on a certain material fail- ure mechanism underlies the assumption that ductile strength of a material has a lognormal distribution. Suppose the parameters are 5 and = .1.a. Compute E(X) and F(X).b. Compute P(X > 125).c. Compute P(110 x125).d. What is the value of
Suppose the proportion X of surface area in a randomly selected quadrat that is covered by a certain plant has a stan- dard beta distribution with a 5 and B =
a. Compute E(X) and V(X).b. Compute P(X.2).c. Compute P(.2 X 4).d. What is the expected proportion of the sampling region not covered by the plant?
The accompanying normal probability plot was constructed from a sample of 30 readings on tension for mesh screens behind the surface of video display tubes used in computer monitors. Does it appear plausible that the tension distribu- tion is normal? x 350 300 250 200 0 = percentile 2
A sample of 15 female collegiate golfers was selected and the clubhead velocity (km/hr) while swinging a driver was determined for each one, resulting in the following data ("Hip Rotational Velocities During the Full Golf Swing." J. of Sports Science and Medicine, 2009: 296-299): 69.0 69.7 72.7
Construct a normal probability plot for the following sam- ple of observations on coating thickness for low-viscosity paint ("Achieving a Target Value for a Manufacturing Process: A Case Study," J. of Quality Technology, 1992:22-26). Would you feel comfortable estimating population mean thickness
The article "A Probabilistic Model of Fracture in Concrete and Size Effects on Fracture Toughness" (Magazine of Con- crete Res., 1996: 311-320) gives arguments for why frac- ture toughness in concrete specimens should have a Weibull distribution and presents several histograms of data that appear
The article "The Load-Life Relationship for M50 Bearings with Silicon Nitride Ceramic Balls" (Lubrication Engr., 1984: 153-159) reports the accompanying data on bearing load life (million revs.) for bearings tested at a 6.45 kN load. 47.1 68.1 68.1 90.8 126.0 146.6 229.0 240.0 103.6 103.6 106.0
94. The accompanying observations are precipitation values dur- ing March over a 30-year period in Minneapolis-St. Paul. .77 1.20 1.74 .47 3.09 3.00 1.62 2.81 2.48 1.31 1.87 .96 .81 1.43 1.51 .32 1.18 1.89 1.20 3.37 2.10 .59 1.35 .90 1.95 2.20 .52 .81 4.75 2.05a. Construct and interpret a normal
Let the ordered sample observations be denoted by YYY (y, being the smallest and y, the largest). Our suggested check for normality is to plot the (((-5)/n), y) pairs. Suppose we believe that the observations come from a distribution with mean 0, and let w, w, be the ordered absolute values of the
The following failure time observations (1000s of hours) resulted from accelerated life testing of 16 integrated circuit chips of a certain type: 82.81 11.6 359.5 502.5 307.8 179.7 242.0 26.5 244.8 304.3 379.1 212.6 229.9 558.9 366.7 204.6 Use the corresponding percentiles of the exponential
A 12-in. bar that is clamped at both ends is to be subjected to an increasing amount of stress until it snaps. Let Y = the distance from the left end at which the break occurs. Suppose Y has pdf f(y) (24) (1-12) 0 y 12 {(2) Compute the following:a. The cdf of Y, and graph it. otherwiseb. P(Y4),
Let X denote the time to failure (in years) of a certain hydraulic component. Suppose the pdf of X is f(x)=32/(x+4) for x >
a. Verify that f(x) is a legitimate pdf.b. Determine the cdf.c. Use the result of part (b) to calculate the probability that time to failure is between 2 and 5 years.d. What is the expected time to failure?e. If the component has a salvage value equal to 100/(4 + x) when its time to failure is x,
The completion time X for a certain task has cdf F(x) given by - -x 1 x
The breakdown voltage of a randomly chosen diode of a certain type is known to be normally distributed with mean value 40 V and standard deviation 1.5 V.a. What is the probability that the voltage of a single diode is between 39 and 42?b. What value is such that only 15% of all diodes have voltages
The article "Characterization of Room Temperature Damping in Aluminum-Indium Alloys" (Metallurgical Trans., 1993: 1611-1619) suggests that Al matrix grain size (um) for an alloy consisting of 2% indium could be modeled with a normal distribution with a mean value 96 and standard deviation
a. What is the probability that grain size exceeds 100?b. What is the probability that grain size is between 50 and 80?c. What interval (a,b) includes the central 90% of all grain sizes (so that 5% are below a and 5% are above b)?
The reaction time (in seconds) to a certain stimulus is a continuous random variable with pdf (x)= 22 otherwisea. Obtain the cdf.b. What is the probability that reaction time is at most 2.5 sec? Between 1.5 and 2.5 sec?e. Compute the expected reaction time.d. Compute the standard deviation of
Let X denote the temperature at which a certain chemical reaction takes place. Suppose that X has pdf (4-x)-1x2 f(x)= = 0 otherwisea. Sketch the graph of f(x).b. Determine the cdf and sketch it.c. Is 0 the median temperature at which the reaction takes place? If not, is the median temperature
What kind of distribution does Y have? (Give the names and values of any parameters.)
The article "The Prediction of Corrosion by Statistical Analysis of Corrosion Profiles" (Corrosion Science, 1985: 305-315) suggests the following cdf for the depth X of the deepest pit in an experiment involving the exposure of carbon manganese steel to acidified seawater. F(x,a, B) e = The authors
Assume this to be the correct model.a. What is the probability that the depth of the deepest pit is at most 150? At most 300? Between 150 and 300?b. Below what value will the depth of the maximum pit be observed in 90% of all such experiments?c. What is the density function of X?d. The density
Let the amount of sales tax a retailer owes the govern- ment for a certain period. The article "Statistical Sampling in Tax Audits" (Statistics and the Law, 2008: 320-343) proposes modeling the uncertainty in 1 by regarding it as a normally distributed random variable with mean value and standard
The proposed penalty (i.e., loss) function for over- or under-assessment is L(a,1) taift>a and = k(at) if tsa (k>1 is suggested to incorporate the idea that over-assessment is more serious than under-assessment).a. Show that a = + -(1/(k + 1)) is the value of a that minimizes the expected loss,
The mode of a continuous distribution is the value x* that maximizes f(x).a. What is the mode of a normal distribution with param- eters and ?b. Does the uniform distribution with parameters A and B have a single mode? Why or why not?c. What is the mode of an exponential distribution with parameter
In some systems, a customer is allocated to one of two service facilities. If the service time for a customer served by facility i has an exponential distribution with parameter A, (i = 1, 2) and p is the proportion of all customers served by facility 1, then the pdf of X = the service time of a
Suppose a particular state allows individuals filing tax returns to itemize deductions only if the total of all item- ized deductions is at least $5000. Let X (in 1000s of dol- lars) be the total of itemized deductions on a randomly chosen form. Assume that X has the pdf Sk/x k/x" x5 fix;a) = {
Let I, be the input current to a transistor and I, be the out- put current. Then the current gain is proportional to In(II). Suppose the constant of proportionality is 1 (which amounts to choosing a particular unit of measure- ment), so that current gain In(I/I). Assume X is normally distributed
The article "Response of SiC/Si,N, Composites Under Static and Cyclic Loading An Experimental and Statistical Analysis" (J. of Engr. Materials and Technology, 1997: 186-193) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull distribution with a
a. Sketch a graph of the density function.b. What is the probability that the strength of a randomly selected specimen will exceed 175? Will be between 150 and 175?c. If two randomly selected specimens are chosen and their strengths are independent of one another, what is the probability that at
Let Z have a standard normal distribution and define a new rv Y by Y=Z+. Show that Y has a normal distribu- tion with parameters and . [Hint: Y yiff Z? Use this to find the cdf of Y and then differentiate it with respect to y.]
a. Suppose the lifetime X of a component, when measured in hours, has a gamma distribution with parameters a and B. Let Y = the lifetime measured in minutes. Derive the pdf of Y. [Hint: Yy iff Xy/60. Use this to obtain the cdf of Y and then differentiate to obtain the pdf.]b. If X has a gamma
Let X denote the lifetime of a component, with f(x) and F(x) the pdf and cdf of X. The probability that the compo- nent fails in the interval (x, x + Ax) is approximately f(x) Ax The conditional probability that it fails in (x,xAx) given that it has lasted at least x is f(x) Ax/[1 F(x)] Dividing
Let U have a uniform distribution on the interval [0, 1]. Then observed values having this distribution can be ob- tained from a computer's random number generator. Let X (1/A)In(1- U).a. Show that X has an exponential distribution with param- eter A. [Hint: The cdf of X is F(x) = P(X x); X x is
Consider an rv X with mean and standard deviation , and let g(X) be a specified function of X. The first-order Taylor series approximation to g(X) in the neighborhood of g(x) = g()+g() (X-p) The right-hand side of this equation is a linear function of X. If the distribution of X is concentrated in
A function g(x) is convex if the chord connecting any two points on the function's graph lies above the graph. When g(x) is differentiable, an equivalent condition is that for every x, the tangent line at x lies entirely on or below the graph. (See the figure below.) How does g(u) = g(E(X)) compare
Let V denote rainfall volume and W denote runoff volume (both in mm). According to the article "Runoff Quality Analysis of Urban Catchments with Analytical Probability Models" (J. of Water Resource Planning and Management, 2006: 4-14), the runoff volume will be 0 if Vv, and will be k(V-va) if V>v.
6. Starting at a fixed time, each car entering an intersection is observed to see whether it turns left (L), right (R), or goes straight ahead (A). The experiment terminates as soon as a car is observed to turn left. Let of cars observed. What are possible X values? List five outcomes and their
5. If the sample space is an infinite set, does this necessarily imply that any rv X defined from will have an infinite set of possible values? If yes, say why. If no, give an example.
4. Let of nonzero digits in a randomly selected zip code. What are the possible values of X? Give three possible outcomes and their associated X values.X 5 the number X 5 the number
1. A concrete beam may fail either by shear (S) or flexure (F).Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let of beams among the three selected that failed by shear. List each outcome in the sample space along with the associated value
114. Show that if A1, A2, and A3 are independent events, then P(A1 . | A2 ¨ A3) 5 P(A1)P(A1) ?P(A2) ?P(A3)P(A1 ¨ A2 ¨ A3) 2 A2 5 5win prize 26 A3 5 5win prize 36 A1 5 5win prize 16 pi 5 P(Ai) for i 5 1,2,3,4 s 5 0 s 5 2
113. A box contains the following four slips of paper, each having exactly the same dimensions: (1) win prize 1; (2) win prize 2; (3) win prize 3; (4) win prizes 1, 2, and 3. One slip will be randomly selected. Let ,, and . Show that A1 and A2 are independent, that A1 and A3 are independent, and
112. Consider four independent events A1, A2, A3, and A4, and let. Express the probability that at least one of these four events occurs in terms of the pis, and do the same for the probability that at least two of the events occur.
111. A personnel manager is to interview four candidates for a job. These are ranked 1, 2, 3, and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously
110. A particular airline has 10 A.M. flights from Chicago to New York, Atlanta, and Los Angeles. Let A denote the event that the New York flight is full and define events B and C analogously for the other two flights. Suppose, , and the three events are independent. What is the probability thata.
108. In a Little League baseball game, team A’s pitcher throws a strike 50% of the time and a ball 50% of the time, successive pitches are independent of one another, and the pitcher never hits a batter. Knowing this, team B’s manager has instructed the first batter not to swing at anything.pi
107. A subject is allowed a sequence of glimpses to detect a target.Let , with . Suppose the Gi ' s are independent events, and write an expression for the probability that the target has been detected by the end of the nth glimpse. [Note:This model is discussed in “Predicting Aircraft
106. One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the sun’s energy reflected from the rock surface.Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths; typically, for granite,
105. Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days.a. If ten people are randomly selected, what is the probability that all have different birthdays? That at least two have the same
103. Refer to Exercise 102. Suppose that 50% of the overnight parcels are sent via express mail service E2 and the remaining 10% are sent via E3. Of those sent via E2, only 1% arrive late, whereas 5% of the parcels handled by E3 arrive late.a. What is the probability that a randomly selected parcel
102. A certain company sends 40% of its overnight mail parcels via express mail service E1. Of these parcels, 2% arrive after the guaranteed delivery time (denote the event “late delivery” by L). If a record of an overnight mailing is randomly selected from the company’s file, what is the
100. One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 90% detection rate for carriers and a 5%detection rate for noncarriers. Suppose the test is applied independently to two different blood samples from the same
99. Fasteners used in aircraft manufacturing are slightly crimped so that they lock enough to avoid loosening during vibration. Suppose that 95% of all fasteners pass an initial inspection. Of the 5% that fail, 20% are so seriously defective that they must be scrapped. The remaining fasteners are
98. Each contestant on a quiz show is asked to specify one of six possible categories from which questions will be asked.Suppose and successive contestants choose their categories independently of one another. If there are three contestants on each show and all three contestants on a particular
96. According to the article “Optimization of Distribution Parameters for Estimating Probability of Crack Detection”(J. of Aircraft, 2009: 2090–2097), the following “Palmberg”equation is commonly used to determine the probability Pd(c) of detecting a crack of size c in an aircraft
95. Individual A has a circle of five close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the five friends to a party to circulate the rumor. To begin, A selects one of the five at random and tells the rumor to the chosen individual. That
94. A transmitter is sending a message by using a binary code, namely, a sequence of 0’s and 1’s. Each transmitted bit (0 or 1) must pass through three relays to reach the receiver. At each relay, the probability is .20 that the bit sent will be different from the bit received (a reversal).
93. One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air Force Base in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral launch goes off
91. A factory uses three production lines to manufacture cans of a certain type. The accompanying table gives percentages of nonconforming cans, categorized by type of nonconformance, for each of the three lines during a particular time period.Line 1 Line 2 Line 3 Blemish 15 12 20 Crack 50 44 40
90. A small manufacturing company will start operating a night shift. There are 20 machinists employed by the company.a. If a night crew consists of 3 machinists, how many different crews are possible?b. If the machinists are ranked 1, 2, . . . , 20 in order of competence, how many of these crews
89. Suppose identical tags are placed on both the left ear and the right ear of a fox. The fox is then let loose for a period of time. Consider the two events and. Let , and assume C1 and C2 are independent events. Derive an expression(involving ) for the probability that exactly one tag is lost,
87. Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A1, A2, and A3 by A3 5 likes vehicle [3 A1 5 likes vehicle [1 A2 5 likes vehicle [2 P(A ¨ B)P(A ¨ B)P(A ¨ B)P(A ¨ B)P(A ¨ B)P(A) 5 P(B) 5 .2 P(A ¨ B)B 5 5the second board
1. A box contains three marbles: one red, one green, and one blue.Consider an experiment that consists of taking one marble from the box then replacing it in the box and drawing a second marble from the box. What is the sample space? If, at all times, each marble in the box is equally likely to be
86.a. A lumber company has just taken delivery on a lot of boards. Suppose that 20% of these boards(2,000) are actually too green to be used in first-quality construction. Two boards are selected at random, one after the other. Let and. Compute P(A), P(B), and (a tree diagram might help). Are A and
*2. Repeat Exercise 1 when the second marble is drawn without replacing the first marble.
3. A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, what is the probability that it will be tossed exactly four times?
4. Let be three events. Find expressions for the events that of(a) only F occurs,(b) both E and F but not G occur,(c) at least one event occurs,(d) at least two events occur,(e) all three events occur,(f) none occurs,(g) at most one occurs,(h) at most two occur.
85. A quality control inspector is inspecting newly produced items for faults. The inspector searches an item for faults in a series of independent fixations, each of a fixed duration.Given that a flaw is actually present, let p denote the probability that the flaw is detected during any one
*5. An individual uses the following gambling system at Las Vegas.He bets $1 that the roulette wheel will come up red. If he wins, he quits. If he loses then he makes the same bet a second time only this time he bets $2; and then regardless of the outcome, quits.Assuming that he has a probability
6. Show that .
7. Show that .
8. If and , show that . In general, show that This is known as Bonferroni's inequality.
*9. We say that if every point in E is also in F. Show that if, then
84. Seventy percent of all vehicles examined at a certain emissions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the following probabilities:a. P(all of the next three vehicles inspected pass)b. P(at least one of the
10. Show that This is known as Boole's inequality.Hint: Either use Eq. (1.2) and mathematical induction, or else show that , where , and use property(iii) of a probability.
11. If two fair dice are tossed, what is the probability that the sum is?
12. Let E and F be mutually exclusive events in the sample space of an experiment. Suppose that the experiment is repeated until either event E or event F occurs. What does the sample space of this new super experiment look like? Show that the probability that event E occurs before event F is
81. Refer back to the series-parallel system configuration introduced in Example 2.35, and suppose that there are only two cells rather than three in each parallel subsystem [in Figure 2.14(a), eliminate cells 3 and 6, and renumber cells 4 and 5 as 3 and 4]. Using , the probability that system
80. Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works iff either 1 or 2 works;since 3 and 4 are connected in series, that subsystem works iff both 3 and 4 work. If components work independently of one

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