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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

a. Under the same conditions as those leading to the interval(7.5), . Use this to derive a one-sided interval for m that has infinite width and provides a lower confidence bound on m. What is this
Let , with . a Then a1 . 0, a2 . 0 1 1 a2 5 a s 5 100 n 5 100 x 5 58.3 n 5 100 x 5 58.3 n 5 100 x 5 58.3 n 5 25 x 5 58.3 s 5 3.0a. Use this equation to derive a more general expression for a CI for m
On the basis of extensive tests, the yield point of a particular type of mild steel-reinforcing bar is known to be normally distributed with . The composition of bars has been slightly modified, but
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation .75.a. Compute a 95% CI for the true average
A CI is desired for the true average stray-load loss m (watts)for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that strayload loss is
Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let m denote the average alcohol content for the
Each of the following is a confidence interval for m true average (i.e., population mean) resonance frequency (Hz) for all tennis rackets of a certain type:(114.4, 115.6) (114.1, 115.9)a. What is the
Consider a normal population distribution with the value of s known.a. What is the confidence level for the interval?b. What is the confidence level for the interval?c. What value of in the CI
The mean squared error of an estimator is. If is unbiased, then but in general MSE( . Consider the esti- ˆu) 5 V( ˆu) 1 (bias)2 MSE(ˆu) 5 V(ˆu), ˆ MSE( u ˆu) 5 E(ˆu 2 u)2ˆu SUPPLEMENTARY
The mean squared error of an estimator is. If is unbiased, then but in general MSE( . Consider the esti- ˆu) 5 V( ˆu) 1 (bias)2 MSE(ˆu) 5 V(ˆu), ˆ MSE( u ˆu) 5 E(ˆu 2 u)2ˆu Use the fact that
At time t 0, there is one individual alive in a certain population. A pure birth process then unfolds as follows. The time until the first birth is exponentially distributed with parameter l. After
a. Let X1, . . . , Xn be a random sample from a uniform distribution on [0, ]. Then the mle of is . ˆu 5 Y 5 max(Xi u u )X P(|Y 2 mY | $ P) # sY 2 /P X m `P u ˆuˆ P(| S ` u ˆu 2 u| $ P) S 0 P ˆu
An estimator is said to be consistent if for any 0, as n . That is, is consistent if, as the sample size gets larger, it is less and less likely that will be further than from the true value of .Show
At time t 0, 20 identical components are tested. The lifetime distribution of each is exponential with parameter l.The experimenter then leaves the test facility unmonitored.On his return 24 hours
Consider a random sample X1, X2, . . . , Xn from the shifted exponential pdf f(x; l, u) 5 e le2l(x2u) x $ u 0 otherwise uu m mˆ 5 X mTaking 0 gives the pdf of the exponential distribution considered
Let X1, X2, . . . , Xn represent a random sample from the Rayleigh distribution with density function given in Exercise 15.Determinea. The maximum likelihood estimator of , and then calculate the
Let X1, . . . , Xn be a random sample from a gamma distribution with parameters a and b.a. Derive the equations whose solutions yield the maximum likelihood estimators of a andb. Do you think they
Refer to Exercise 25.Suppose we decide to examine another test spot weld. Let X shear strength of the weld.Use the given data to obtain the mle of P(X 400). [Hint:P(X 400) ((400 )/s).]
The shear strength of each of ten test spot welds is determined, yielding the following data (psi):392 376 401 367 389 362 409 415 358 375a. Assuming that shear strength is normally distributed,
Two different computer systems are monitored for a total of n weeks. Let Xi denote the number of breakdowns of the first system during the ith week, and suppose the Xi’s are independent and drawn
Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test.Suppose the pdf of X is where 1 . A random sample of ten students yields data
Let X have a Weibull distribution with parameters a andb, so E(X) b (1 1/a)V(X) b2{(1 2/a) [(1 1/a)]2}a. Based on a random sample X1, . . . , Xn, write equations for the method of moments
An investigator wishes to estimate the proportion of students at a certain university who have violated the honor code. Having obtained a random sample of n students, she realizes that asking each,
Let X1, X2, . . . , Xn be a random sample from a pdf f(x) that is symmetric about m, so that is an unbiased estimator of. If n is large, it can be shown that V( ) 1/(4n[ f(m)]2).a. Compare V( ) to V(
In Chapter 3, we defined a negative binomial rv as the number of failures that occur before the rth success in a sequence of independent and identical success/failure trials.The probability mass
Suppose the true average growth of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is s2, whereas for the second type the
Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdfa. It can be shown that E(X2) 2 . Use this fact to construct an unbiased estimator of based on (and use rules of
A sample of n captured Pandemonium jet fighters results in serial numbers x1, x2, x3, . . . , xn. The CIA knows that the aircraft were numbered consecutively at the factory starting with a and ending
Consider a random sample X1, . . . , Xn from the pdf f(x; ) .5(1 x) 1 x 1 where 1 1 (this distribution arises in particle physics). Show that is an unbiased estimator of .[Hint: First determine
Suppose a certain type of fertilizer has an expected yield per acre of 1 with variance s2, whereas the expected yield for a second type of fertilizer is 2 with the same variance s2.Let and denote the
Of n1 randomly selected male smokers, X1 smoked filter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked filter cigarettes. Let p1 and p2 denote the probabilities that a randomly
Using a long rod that has length , you are going to lay out a square plot in which the length of each side is . Thus the area of the plot will be 2. However, you do not know the value of , so you
Each of 150 newly manufactured items is examined and the number of scratches per item is recorded (the items are supposed to be free of scratches), yielding the following data:Number of scratches per
In a random sample of 80 components of a certain type, 12 are found to be defective.a. Give a point estimate of the proportion of all such components that are not defective.b. A system is to be
a. A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected and the amount of gas (therms) used during the month of January is determined for each
Consider the accompanying observations on stream flow(1000s of acre-feet) recorded at a station in Colorado for the period April 1–August 31 over a 31-year span (from an article in the 1974 volume
As an example of a situation in which several different statistics could reasonably be used to calculate a point estimate, consider a population of N invoices. Associated with each invoice is its
Calculate the estimate for the given data.b. Use rules of variance from Chapter 5 to obtain an expression for the variance and standard deviation (standard error) of the estimator in part (a), and
The article from which the data in Exercise 1 was extracted also gave the accompanying strength observations for cylinders:6.1 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.8 8.1 7.4 8.5 8.9 9.8 9.7 14.1
Consider the following sample 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:
The accompanying data on flexural strength (MPa) for concrete beams of a certain type was introduced in Example 1.2.5.9 7.2 7.3 6.3 8.1 6.8 7.0 7.6 6.8 6.5 7.0 6.3 7.9 9.0 8.2 8.7 7.8 9.7 7.4 7.7 9.7
A more accurate approximation to E[h(X1, . . . , Xn)] in Exercise 93 is h(m1, c,mn) 1 12 s1 2a'2 h'x1 2 b 1 c1 12 sn 2a'2 h'xn 2b Y 5 X4c 1 X1 11 X2 11 X3 dCompute this for Y h(X1, X2, X3, X4) given
Let A denote the percentage of one constituent in a randomly selected rock specimen, and let B denote the percentage of a second constituent in that same specimen.Suppose D and E are measurement
A rock specimen from a particular area is randomly selected and weighed two different times. Let W denote the actual weight and X1 and X2 the two measured weights. Then X1 W E1 and X2 W E2, where E1
a. Let X1 have a chi-squared distribution with parameter n1 (see Section 4.4), and let X2 be independent of X1 and have a chi-squared distribution with parameter n2. Use the technique of Example 5.21
a. Use the general formula for the variance of a linear combination to write an expression for V(aX Y). Then let a sY /sX, and show that r 1. [Hint: Variance is always 0, and Cov(X, Y ) sX sY
We have seen that if E(X1) E(X2) ... E(Xn) m, then E(X1 ... Xn) nm. In some applications, the number of Xi’s under consideration is not a fixed number n but instead is an rv N. For example, let N
Suppose that for a certain individual, calorie intake at breakfast is a random variable with expected value 500 and standard deviation 50, calorie intake at lunch is c (a 1 bt) >(c 1 dt 1 et2)1/2d
In Exercise 66, the weight of the beam itself contributes to the bending moment. Assume that the beam is of uniform thickness and density so that the resulting load is uniformly distributed on the
If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at 0 due to the loads is a1X1 a2X2.c. With Xi denoting the number of cars entering from road i
Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the number of cars coming from each road onto the freeway is a random variable, with expected
Suppose that when the pH of a certain chemical compound is 5.00, the pH measured by a randomly selected beginning chemistry student is a random variable with mean 5.00 and standard deviation .2. A
Five automobiles of the same type are to be driven on a 300-mile trip. The first two will use an economy brand of gasoline, and the other three will use a name brand. Let X1, X2, X3, X4, and X5 be
Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility.Suppose they are independent, normal rv’s with expected values m1, m2, and m3
There are two traffic lights on a commuter’s route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he
a. Compute the covariance between X and Y in Exercise 9.b. Compute the correlation coefficient r for this X and Y.fY ( y) 5 e 2y 0 # y # 1 0 otherwise fX(x) 5 e 3x2 0 # x # 1 0 otherwise
Annie and Alvie have agreed to meet for lunch between noon (0:00 P.M.) and 1:00 P.M. Denote Annie’s arrival time by X, Alvie’s by Y, and suppose X and Y are independent with pdf’s What is the
Consider a system consisting of three components as pictured. The system will continue to function as long as the f(x, y) 5 e xe2x(11y) x $ 0 and y $ 0 0 otherwise gm k50 am kbak bm2k 5 (a 1 b)m
Two different professors have just submitted final exams for duplication. Let X denote the number of typographical errors on the first professor’s exam and Y denote the number of such errors on the
Let X denote the number of Canon digital cameras sold during a particular week by a certain store. The pmf of X is Sixty percent of all customers who purchase these cameras also buy an extended
A service station has both self-service and full-service islands.On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the
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
An individual’s credit score is a number calculated based on that person’s credit history that helps a lender determine how much he/she should be loaned or what credit limit should be established
Let X have a Weibull distribution with parameters and . Show that has a chi-squared distribution with . [Hint: The cdf of Y is ; express this probability in the form , use the fact that X has a cdf
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
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 is The right-hand side of
Let U have a uniform distribution on the interval [0, 1].Then observed values having this distribution can be obtained from a computer’s random number generator. Let X 5 2(1/l)ln(1 2 U).b r(x) 5
Let X denote the lifetime of a component, with f(x) and F(x) the pdf and cdf of X. The probability that the component fails in the interval is approximately. The conditional probability that it fails
The article “Three Sisters Give Birth on the Same Day”(Chance, Spring 2001, 23–25) used the fact that three Utah sisters had all given birth on March 11, 1998 as a basis for s 5 20 f(v) 5 v s2
In Exercises 117 and 118, as well as many other situations, one has the pdf f(x) of X and wishes to know the pdf of. Assume that is an invertible function, so that can be solved for x to yield . Then
a. Suppose the lifetime X of a component, when measured in hours, has a gamma distribution with parameters and . Let the lifetime measured in minutes.Derive the pdf of Y. [Hint: iff . Use this to
Let Z have a standard normal distribution and define a new rv Y by . Show that Y has a normal distribution with parameters and . [Hint: iff ? Use this to find the cdf of Y and then differentiate it
Let be the input current to a transistor and be the output current. Then the current gain is proportional to. Suppose the constant of proportionality is 1(which amounts to choosing a particular unit
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 P(X # 0), P(X # 2),
The article “Error Distribution in Navigation” (J. of the Institute of Navigation, 1971: 429–442) suggests that the frequency distribution of positive errors (magnitudes of errors) is well
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
The reaction time (in seconds) to a certain stimulus is a continuous random variable with pdfa. Obtain the cdf.b. What is the probability that reaction time is at most 2.5 sec? Between 1.5 and 2.5
The completion time X for a certain task has cdf F(x) given bya. Obtain the pdf f(x) and sketch its graph.b. Compute .c. Compute E(X).
Let X denote the time to failure (in years) of a certain hydraulic component. Suppose the pdf of X is for .a. Verify that f(x) is a legitimate pdf.b. Determine the cdf.c. Use the result of part (b)
A 12-in. bar that is clamped at both ends is to be subjected to an increasing amount of stress until it snaps. Let the distance from the left end at which the break occurs.Suppose Y has pdf Compute
Let the time it takes a read/write head to locate a desired record on a computer disk memory device once the head has been positioned over the correct track. If the disks rotate once every 25
Let the ordered sample observations be denoted by( being the smallest and the largest). Our suggested check for normality is to plot the pairs. Suppose we believe that the observations come from a
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
Stress is applied to a 20-in. steel bar that is clamped in a fixed position at each end. Let the distance from the left end at which the bar snaps. Suppose Y/20 has a standard beta distribution with
Let X have a standard beta density with parameters and .a. Verify the formula for E(X) given in the section.b. Compute . If X represents the proportion of a substance consisting of a particular
Suppose the proportion X of surface area in a randomly selected quadrat that is covered by a certain plant has a standard beta distribution with and .a. Compute E(X) and V(X).b. Compute .c. Compute
The article “The Statistics of Phytotoxic Air Pollutants”(J. of Royal Stat. Soc., 1989: 183–198) suggests the lognormal distribution as a model for concentration above a certain forest. Suppose
A theoretical justification based on a certain material failure mechanism underlies the assumption that ductile strength X of a material has a lognormal distribution.Suppose the parameters are and
a. Use Equation (4.13) to write a formula for the median of the lognormal distribution. What is the median for the load distribution of Exercise 79?b. Recalling that is our notation for the
The article “On Assessing the Accuracy of Offshore Wind Turbine Reliability-Based Design Loads from the Environmental Contour Method” (Intl. J. of Offshore and Polar Engr., 2005: 132–140)
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 and .a. What are the
Let X have a Weibull distribution with the pdf from Expression (4.11). Verify that . [Hint: In the integral for E(X), make the change of variable, so that .]
Let the time (in weeks) from shipment of a defective product until the customer returns the product. Suppose that the minimum return time is and that the excess g 5 3.5 1021 X 5 a 5 2.5 b 5 200 P(1.5
The lifetime X (in hundreds of hours) of a certain type of vacuum tube has a Weibull distribution with parameters and . Compute the following:a. E(X) and V(X)b.c.(This Weibull distribution is
a. The event { } is equivalent to what event involving X itself?b. If X has a standard normal distribution, use part (a) to write the integral that equals . Then differentiate this with respect to y
A system consists of five identical components connected in series as shown:F(t; l, n) 5 P(X # t)1 2345 As soon as one component fails, the entire system will fail.Suppose each component has a
The special case of the gamma distribution in which is a positive integer n is called an Erlang distribution. If we replace by in Expression (4.8), the Erlang pdf is It can be shown that if the times
Let X have a standard gamma distribution with .Evaluate the following:a.b. c.d. e.f.
Evaluate the following:a. (6)b. (5/2)c. F(4; 5) (the incomplete gamma function)d. F(5; 4)e. F(0 ; 4)
A consumer is trying to decide between two long-distance calling plans. The first one charges a flat rate of per minute, whereas the second charges a flat rate of for calls up to 20 minutes in

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