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introduction to probability statistics

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

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 interval for the data in Exercise 5(a)?b. Generalize the result of part (a) to obtain a lower bound
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 of which the interval (7.5) is a special case.b. Let and . Does this result in a narrower or wider
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 the modification is not believed to have affected either the normality or the value of s.a. Assuming
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 porosity of a certain seam if the average porosity for 20 specimens from the seam was 4.85.b. Compute a
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 normally distributed with .a. Compute a 95% CI for m when and .b. Compute a 95% CI for m when and .c.
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 population of all bottles of the brand under study.Suppose that the resulting 95% confidence interval is
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 value of the sample mean resonance frequency?b. Both intervals were calculated from the same sample
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 formula (7.5) results in a confidence level of 99.7%?d. Answer the question posed in part (c) for a
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 EXERCISES (31–38)Use the fact that Y y iff each Xi y to derive the cdf of Y. Then show that the pdf of
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 Y y iff each Xi y to derive the cdf of Y. Then show that the pdf of Y max(Xi) isb. Use the result of
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 the first birth, there are two individuals alive. The time until the first gives birth again is
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 that is a consistent estimator of when s2 by using Chebyshev’s inequality from Exercise 44 of
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 later, the experimenter immediately terminates the test after noticing that y 15 of the 20 components
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 previously (with positive density to the right of zero). An example of the shifted exponential
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 estimate for the vibratory stress data given in that exercise. Is this estimator the same as 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 can be solved explicitly?b. Show that the mle of ab is .
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, estimate the true average shear strength and standard deviation of shear strength using the method of
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 from a Poisson distribution with parameter m1. Similarly, let Yi denote the number of breakdowns of
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 x1 .92, x2 .79, x3 .90, x4 .65, x5 .86, x6 .47, x7 .73, x8 .97, x9 .94, x10 .77.u f(x; u) 5 e(u 1
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 estimators of b anda. Show that, once the estimate of a has been obtained, the estimate of b can be
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, “Have you violated the honor code?” will probably result in some untruthful responses.Consider
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( ) when the underlying distribution is normal.b. When the underlying pdf is Cauchy (see Example
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 function (pmf) of X is nb(x; r, p)a x 1 r 2 1 x b pr(1 2 p)x x 5 0, 1, 2, .a. Suppose that r 2.Show
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 variance is 4s2. Let X1, . . . , Xm be m independent growth observations on the first type [so
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 expected value to show that it is unbiased).b. Estimate from the following n 10 observations on
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 withb, so that the total number of planes manufactured is b a 1 (e.g., if a 17 and b 29, then 29
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 E(X) E( ).]
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 sample variances of yields based on sample sizes n1 and n2, respectively, of the two
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 selected male and female, respectively, smoke filter cigarettes.a. Show that (X1/n1) (X2/n2) is an
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 decide to make n independent measurements X1, X2, . . . , Xn of the length. Assume that each Xi has
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 item 0 1 2 3 4 567 Observed frequency 18 37 42 30 13 7 2 1 Let X the number of scratches on a
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 constructed by randomly selecting two of these components and connecting them in series, as shown here.m
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 house. The resulting observations are 103, 156, 118, 89, 125, 147, 122, 109, 13899.Let denote the
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 of Water Resources Research).An appropriate probability plot supports the use of the lognormal
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 “book value,” the recorded amount of that invoice. Let T denote the total book value, a known
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 then compute the estimated standard error.c. Calculate a point estimate of the ratio s1/s2 of the two
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 12.6 11.2 Prior to obtaining data, denote the beam strengths by X1,..., Xm and the cylinder strengths
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: 22–26):.83 .88 .88 1.04 1.09 1.12 1.29 1.31 1.48 1.49 1.59 1.62 1.65 1.71 1.76 1.83 Assume that the
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 7.8 7.7 11.6 11.3 11.8 10.7a. Calculate a point estimate of the mean value of strength for the
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 in Exercise 93, and compare it to the leading term h(m1, . . . , mn).
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 errors in determining the values of A and B so that measured values are X A D and Y B E, respectively.
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 and E2 are the two measurement errors. Suppose that the Ei s are independent of one another and of W
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 to show that X1 X2 has a chi-squared distribution with parameter n1 n2.f(x, y) 5 •2 5(2x 1 3y) 0
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 r.]b. By considering V(aX Y ), conclude that r 1.c. Use the fact that V(W) 0 only if W is a
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 the number of components that are brought into a repair shop on a particular day, and let Xi denote
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 f(x, y) 5 e kxy x $ 0, y $ 0, 20 # x 1 y # 30 0 otherwise random with expected value 900 and standard
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 beam. If the weight of the beam is random, the resulting load from the weight is also random;denote
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 during the period, suppose that Cov(X1, X2) 80, Cov(X1, X3) 90, and Cov(X2, X3) 100 (so that the
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 value and standard deviation as given in the table.Road 1 Road 2 Road 3 Expected value 800 1000 600
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 large batch of the compound is subdivided and a sample given to each student in a morning lab and each
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 the observed fuel efficiencies (mpg) for the five cars. Suppose these variables are independent and
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 and variances , and , respectively.a. If m m2 m3 60 and , calculate P(To 200) and P(150 To
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 must stop when returning from work.Suppose these two variables are independent, each with pmf given in
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 expected amount of time that the one who arrives first must wait for the other person? [Hint: h(X,
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 first component functions and either component 2 or component 3 functions. Let X1, X2, and X3 denote the
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 second exam. Suppose X has a Poisson A 5 E(x, y): | x 2 y | # 1 6 F f(x, y) 5 e K(x2 1
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 warranty. Let Y denote the number of purchasers during this week who buy an extended warranty.a. What is
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 self-service island at a particular time, and let Y denote the number of hoses on the full-service island in
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 and will be if . Here is
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 for a credit card. An article in the Los Angeles Times gave data which suggested that a beta
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 of the form in Expression (4.12), and differentiate with respect to y to obtain the pdf of Y.]
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 compare to E(g(X))?
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 this equation is a linear function of X. If the distribution of X is concentrated in an interval over
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 • aa1 2 x bb 0 # x # ba. Show that X has an exponential distribution with parameter . [Hint: The cdf
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 in given that it has lasted at least x is. Dividing this by produces the failure rate function:An
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 # e2v2/2s2 v . 0 s2 Y 5 2ln(X)A 5 0 B 5 1 g(y) 5 f[k(y)] # |kr(y)|y 5 h(x) x 5 k(y)h( # y 5 h(X) )Y
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 it can be shown that the pdf of Y isa. If X has a uniform distribution with and , derive the pdf of
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 obtain the cdf of Y and then differentiate to obtain the pdf.]b. If X has a gamma distribution with
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 with respect to y.]
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 of measurement), so that current gain . Assume X is normally distributed with and .a. What type of
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), P(21 # X # 2)f(x) 5 (.1)e2.2|x| 2` , x , `X 5 na . 1 ba lsm m21 a* 5 m 1 s21(1/(k 1 1))5 5 mba E(X) <
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 approximated by an exponential distribution.Let the lateral position error (nautical miles), which can
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.The authors propose
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 sec?f(x) 5 •3 2# 1 x2 1 # x # 3 0 otherwise(mm)P(.5 # X # 2)c. Compute the expected reaction time.d.
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) to calculate the probability that time to failure is between 2 and 5 years.d. What is the expected
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 the following:a. The cdf of Y, and graph it.b. , andc. E(Y), , and V(Y)d. The probability that the
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 millisec, a reasonable assumption is that X is uniformly distributed on the interval [0, 25].a. Compute .b.
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 distribution with mean 0, and let be the ordered absolute values of the .A half-normal plot is 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 Velocities During the Full Golf Swing,”J. of Sports Science and Medicine, 2009: 296–299):Construct a
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 and .a. What are the parameters of the relevant standard beta distribution?b. Compute .c. Compute
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 ingredient, what is the expected proportion that does not consist of this ingredient?
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 .d. What is the expected proportion of the sampling region not covered by the plant?
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 the parameter values are and .a. What are the mean value and standard deviation of concentration?b.
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. Compute E(X) and V(X).m 5 5 s 5 .1 100(1 2 a)z 100(1 2a) a m|CV 5 .40 (CV 5 sX/mX)X 5 0 , p , 1 a
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 percentile of the standard normal distribution, write an expression for the percentile of the lognormal
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) proposes the Weibull distribution with and as a model for 1-hour significant wave height (m) at a certain
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 mean value and standard deviation of lifetime?b. What is the probability that lifetime is at most
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 # X # 6)P(X # 6)a 5 2 b 5 3 over the minimum has a Weibull distribution with parameters and (see
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 suggested as a model for time in service in “On the Assessment of Equipment Reliability:Trading Data
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 to obtain the pdf of [the square of a N(0, 1) variable]. Finally, show that has a chi-squared
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 lifetime that is exponentially distributed with and that components fail independently of one another.
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 between successive events are independent, each with an exponential distribution with parameter ,
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 duration and then for each additional minute exceeding 20 (assume that calls lasting a noninteger number

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