Their sampling distributions are Normal. Thus, for large n, ML estimators are optimal and their statistical uncertainty can always be assessed. For small sample sizes n the above properties become looser approximations to the true behavior of these estimators. In particular, the true sampling distribution becomes skewed for small n, and the estimators become somewhat biased, although it is sometimes possible to determine a function b n that corrects that bias. An additional important property of ML estimators is that of invariance under transformation.
The terms Cov? See Example When the expected information matrix 3. That is, the ML parameter estimates are substituted directly into the elements of the above matrix, without the need to perform the expectation operation. The local matrix 3. Many statisticians prefer to use the local information matrix, even when the expected matrix can be obtained. For the models discussed in this text, both versions are presented, when they differ. Sometimes, data samples in engineering are censored. That is, not all observations in the sample are actually known.
See Section However, the likelihood function of censored samples is easily expressed, and the ML method proceeds with only minor computational alterations. Compare, for example, the ML equations The ML method of estimation is often the only feasible option for censored samples. Because the assessment of the uncertainty associated with estimated decision functions g 8 is of crucial importance in engineering, estimators for which sampling distributions are not at all available should not be used under any circumstances, when a sample of observations is available.
Because of their asymptotically optimal properties, and the universality of their application, ML estimators are the recommended choice for engineering statistics. This text uses the ML estimation method. ML estimators, or estimating equations, are given and illustrated for the distribution models discussed. For models that are often applied to phenomena for which censored samples are common, corresponding results are presented for censoring.
The required interval is developed along the following line of reasoning. Before an experiment is conducted, the estimator T of the parameter 8 is a random data function with a sampling pdf f t. For example, suppose one were to sample from a Normal process with known variance a:. An equal-sided 1 - a interval tl,t2 from a sampling pdf f t. The probability 3. Expression 3. Thus, 1 - a is a measure of the statistical assurance that a specific interval estimate covers the unknown parameter 8. The value 1 - a is specified by the decision maker. Clearly, the higher the required confidence level, the wider will be the corresponding interval 11, When the exact sampling distribution of T is known, exact confidence intervals are obtained as in the preceding section.
This is possible for only some statistics and for a few of the models discussed in this text; these cases are presented in corresponding chapters. Usually, exact sampling distributions are not known and approximate methods must be used. This is particularly true for estimating functions of several parameters e. Approximate confidence intervals on 8 can therefore be constructed directly from the Normal pdf see Section Also, the invariance property of ML estimators can be used to construct similar confidence intervals on a decision function g 8.
For example, exact sampling distributions for estimators of Gamma parameters are not available. To construct approximate confidence intervals on these parameters, their ML estimates are first obtained from a sample of observations, with The covariance matrix The diagonal terms of that matrix give the approximate sampling variances of these estimators.
Approximate confidence intervals on the parameters are then obtained from the Normal sampling distribution with mean value equal to the estimated parameter and variance equal to the estimated MVB. For large sample sizes these approximate intervals are accurate. For small to moderate sized samples these approximations are less accurate. An alternative approximate approach, which is more difficult to evaluate but produces more accurate results compared to the Normal approach, is available.
This approach is based on the fact that the following ratio of likelihoods is asymptotically distributed as a Chi-squared variable see Section The likelihood ratio function for a sample. The likelihood ratio approaches its x --distribution more rapidly with increasing sample size n than the ML estimator 6 approaches its Normal distribution.
The likelihood-ratio function therefore provides more accurate confidence intervals for small to moderate sample sizes. Refer to Section When the parameter 0 is multivalued, all parameter components in LR 6 are expressed in terms of the component in question by their ML equations, illustrated as follows: Continuing the preceding example, we have that the likelihood function of a Gamma sample is given by Substituting in the above expression for a its ML equation 1 3.
The confidence interval endpoints 1, and I;! The process of calculating such intervals is somewhat involved. However, a modern computational aid such as Mathcad reduces that process to a simple evaluation of an equation solver. The following chapters present and illustrate those cases for which useful results can be obtained by the likelihood-ratio method. TESTS 3. Rather, the objective is to determine if a given parameter value characterizes the measurement process.
If sample information indicates that this given parameter value is reasonable, a certain action follows. If that value is not reasonable, a different action is taken. The statistical inference that leads to such a decision is called a test. For example, suppose that the critical dimension of a mass-produced shaft is the diameter of an interference fit with a bearing, with nominal value xd. As the edge of the cutting tool wears during production, the diameters X produced by the lathe increase over time, and at some point the engineer wants to know if the production average p is still acceptable.
If, however, the sample indicates that this proposition is not supported by the data, he replaces the cutting tool. This hypothesis usually represents the existing condition of the investigated phenomenon measured by X. The statistical test decides on H o by evaluating a suitable statistic T from a sample of observations on X. The difficulty here is obviously that the value t produced by a sample will include sampling variation, so that the decision on H o may be in error.
In particular, even when H o is true, the decision may be to reject H o , resulting in an error. This error is called type I, and its probability of occurrence a is controlled at an acceptably low level as follows. A significance test on a hypothesis H o , at the significance level a. The statistical variation of the data function T is modeled by its sampling distribution f t , given that H o is true. From that distribution a critical value t, is chosen so that the type I error probability is kept at an acceptable significance level a.
This critical value then establishes the decision rule: "if t 3 t,, reject H o at the significance level a. Continuing with the preceding example, we suppose that the machined diameter follows a Normal process, set at the mean value x d. In Section In distinction to a significance test, this type of test is called a hypothesis test.
Again, a suitable statistic T , evaluated from a given sample of observations, would lead to a decision: Either reject Ho, implying acceptance of H a , or vice versa. As before, the former decision may be in error with probability a , the choice of which determines the critical test value tc. However, the latter decision of accepting Ho, and therefore rejecting Ha, may also be in error when Ha is in fact true. This error is termed type With tc determined by 3.
This function is called the operating characteristic of the test and needs to be evaluated for a proposed hypothesis test to check if the type I1 error probability is sufficiently small. The complementary probability 1- j3 is called the power of the test to discriminate between the competing hypotheses. If j3 is too large, either the sample size n needs to be increased or the alternative H a needs to be separated further from Ho. For example, suppose that a research engineer at a materials manufacturer has come up with a process modification that promises to improve the mean strength p o of a manufactured material.
A test batch has been produced in the lab and n tc Figure 3. A typical test on two hypotheses. The question of concern would be: "Has the modification improved the mean strength significantly? If the decision is Ho, the process is left alone. If the decision is Ha,the process is changed. To establish a hypothesis test, one needs the sampling distribution of a suitable statistic T under the hypotheses Hoand under Ha. Only a few practical cases are available for which the exact sampling distributions are known.
These are mainly tests on the mean and variance of Normal, Log-Normal, and Exponential variables. Details on these tests are provided in the chapters that cover these models. For other distributions and more interesting parameter functions, the asymptotic Normal properties of ML estimators, or the likelihood ratio statistic, can be used to construct approximate hypothesis tests. There are several measures of discrepancy between the estimated model and the sample distribution, for which the sampling distribution is known, at least approximately. The reason for choosing A is that it tends to be more powerful than competing statistics T in detecting discrepancies in the distribution tails.
Furthermore, critical test values t, are available2 for the practical case when the parameters 6' are estimated from the same data as are used for the test-of-fit. This statistic can in most cases be modified to account for the effect of the sample size n, so that only a small set of critical test values is required. In addition, the test statistic A is simple to evaluate with a modern computational tool like Mathcad. Details are provided for the models covered in this text, when results are available.
See, for example, R. D'Agostino, M. However, a single test measure, such as the statistic 3. In engineering the tail-fit is often of overriding importance. Hence, the engineer routinely checks the model's fit to the data graphically, in addition to doing a statistical significance test. This graphical technique is called probability plotting.
Since the human eye can more easily judge how a straight line, rather than some curved graph, fits data points, the scales of the graph are chosen to produce a linear plot of both the model and the data. The definition of the sample cdf for the data plot is controversial, as many choices exist. However, since probability plotting should only be used for a visual check on the model fit, any one choice called a plotting position is about as good as any other.
Probability distributions in reliability evaluation
The approximation is good to one digit in the third decimal, even for small n. When the model cdf is of closed form, its linearization is often straightforward, resulting in a relation of the form Here Fi stands for the estimated model Fo qi,; g and the functions g are suitable transformations. Plotting gl Fi versus g4 qi gives a straight line for the estimated model, and the data plot gl pi versus g4 x , will follow this straight line approximately if Fo models the data well.
For example, the Weibull cdf Two versions of a probability plot. When the model Fo is not of closed form, the plot can be linearized numerically. For location-scale models see Section 9. For an illustration, see Section This is sometimes done to explore several model candidates, before the estimation of the chosen candidate is undertaken. It is then tempting to estimate two parameters 0 from the intercept g2 6 and the slope g3 0 of a straight line that is somehow fit to the data.
Since the choice of plotting position is subjective, and the sampling distributions of the resulting estimators are not known, this temptation should be resisted, particularly because defensible ML estimates can be computed easily. Although probability plotting does not yield quantitative inferences, relying as it does on the judgment of the viewer, valuable information is sometimes gleaned from the plot. Plot a indicates that the measurements x came from two distinct populations.
Plot b suggests that the true distribution has a longer upper tail i. Plot c suggests that the true distribution is more skewed to the left than Fo. Plot d indicates the presence of a location parameter. In practice, of course, probability plots do not look as "clean" as in Fig. That is, sampling variation will scatter the data points, particularly for small samples. Hence, one needs experience with these plots to build up a basis for judgment.
That is, not all observations of the sample are known. This means that the true rankings of some of the available observations are not known either, and so the correct plotting positions 3. Thus, the order q includes the number of items censored before x ,, occurred. The product limit 3. Its standard error is approximately Kaplan, E. Deviant probability plots. Formula 3. When there is no censoring, 3. A design modification was made to the turbine runner to improve the turbine efficiency at part load. The following measurements were taken over a period of time at a certain wicket-gate setting partially closed : Effective hydraulic Flow rate head H ft Q cfs That is, the measured variable X only admits discrete values x, usually integers as in the case of a counting process.
Examples of counting processes are: the number of components failing during a system's mission, the number of defectives in a manufactured lot, the number of production orders received at a factory, the number of earthquakes experienced in a region, the number of vehicles crossing a bridge, and the number of industrial accidents in a factory. The uncertainty in the number of counts is measured by probability and is modeled by a mathematical function p x. This function gives the probability of occurrence of discrete values x measured in a sample space S: see also Section 1.
The function p is termed the probability mass function pmf , since probabilities are lumped at discrete values x. The two functions are connected via see Fig. The engineering interpretation of these functions is that they are relative frequencies of occurrence of x in a sequence of measurements on X. The discrete probability distributions covered in this text are indexed by a parameter 8 that takes values in a parameter space Q: The parameter 8 can have several components.
The more components there are, the more flexible is the model p x; 8 in fitting discrete data. Although for continuous distribution models some of the parameters 8 relate to the variable X in a consistent way see Chapter 9 , this is not the case for the discrete models discussed in this text. Thus, there are no location parameters, and the scale of the model is not associated with a specific parameter component.
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All parameter components influence, in combination, both the scale and the shape of the model. For example, on inspection a randomly chosen product specimen may or may not prove to be defective. In a performance test a specimen device may or may not meet specifications. During its service life a structure may or may not be exposed to an earthquake of a certain magnitude. A development project may or may not exceed its budget. Typically there is a sequence of occasions, or trials, at which the event in question may or may not occur. If the following conditions hold, the sequence is termed a Bernoulli sequence, after the seventeenth-century mathematician Jakob Bernoulli.
The conditions are: 1. The trials are statistically identical, each resulting in one of only two possible outcomes: the occurrence or nonoccurrence of a specified event. For each trial the probability p of the event's occurrence is constant. The trials are statistically independent.
It is customary to label the occurrence of the event in question as a "success" s and its nonoccurrence as a "failure" f. This sequence of events is a basic experimental scheme that underlies the discrete distribution models discussed in this text. This translates initially to checking the above conditions on which the Bernoulli process is based. That the first condition holds is usually obvious from the problem context: There are only two possible outcomes for each trial, and trials are conducted in identical circumstances.
The second and third conditions can sometimes be affirmed directly; the classical example is the repeated toss of a coin. In many practical instances, however, these conditions are known to not hold precisely, in which case the Bernoulli scheme may provide an approximate model. In particular, the probability p may not be precisely constant from trial to trial because the experiment is not stable over time.
As well, the trials may not be entirely independent because experimental conditions may drift over time in a particular direction. For example, there are many reasons why the characteristics of a machining process change consistently over time: cutting edges dull, gear teeth wear down, bearings loosen, sliding surfaces fret, and so forth. The result is that the probability pof producing a defective item cannot be constant over time, and that the change is likely in a consistent direction, implying the statistical dependence of successive trials.
Nevertheless, the Bernoulli sequence provides a useful standard against which actual process performance can be measured to discover process changes that would call for timely corrective action. When it is not clear whether conditions 2 and 3 hold for the particular discrete sample on hand, these conditions need to be tested. The following simple, but not too powerful, randomness test checks the number of runs observed in the data sequence.
This test is "distribution-free," in that it does not depend on the choice of a particular distribution for modeling the random variable. A run is defined as a subsequence of consecutive outcomes of one kind, immediately preceded and followed by outcomes of the other kind. Thus, in the preceding section the sample sequence shows five runs: f f, s , f f f, s s , and f f f f.
Clearly, to obtain only one run outcomes of one kind only would be viewed as an unusual clustering and would intuitively be taken as evidence against the sample being random.
Similarly, if n runs are found outcomes alternate one by one , improbable mixing would be suspected, and the sample's randomness would also be questioned. Consider a sample sequence with a successes and b failures. If the randomness condition holds, all arrangements of elements in the sequence are equally likely. If clustering is suspected, r would be substantially smaller than the expected value 4.
A test at the significance level a requires the quantile rq such that q equals a, or falls just below a. Swed, C. Eisenhart, Annals o f Mathematical Statistics, Vol. Hence, if r 5 r,, the randomness hypothesis is rejected at the significance level a. The expected value 4.
Thus there appears no reason to reject the randomnes hypothesis. If mixing is suspected, r would be substantially larger than the expected value 4. Note that Tables 4. Elsenhart, Annals of Mathematrcal Stattstrcs, Vol. See Example 4. One-sided critical values, for a significance level a , are for suspected clustering and for suspected mixing, where z, is the standard Normal quantile of order a.
For example, the number of defective items found in a sample from a fixed production lot would be modeled by a Hypergeometric distribution. However, the number of defectives in a sample from a continuous production process would be modeled by a Binomial distribution. The Negative Binomial distribution Chapter 7 models the "inverse sampling" aspect of Bernoulli trials: the number of trials to the rth success.
A special case of the Hypergeometric is the Geometric distribution, which models the number of trials to the first success. For example, a flood-control structure is designed to contain a daily flow volume D. Each year there will be a maximum daily flow volume "annual flood" that may or may not exceed D. The Geometric distribution models the number of years to the first exceedance of D. The Poisson distribution Chapter 8 models situations where an event may or may not occur at any point in time or space.
By dividing the time or space continuum into small intervals such that only one event may occur in any one interval, a Bernoulli sequence of trials results, provided that the probability of occurrence of the event is constant for each trial. Thus, the Poisson distribution is the limiting case of the Binomial distribution, as the number of trials becomes large while the average occurrence rate remains constant.
For example, cracks may occur anywhere along a line of weld. Dividing the length of weld into very small segments trials results in a large number of segments to cover the weld length. If the rate of crack occurrence per unit length is constant, the Poisson distribution models the number of welds occurring along the weld line. In discrete-distribution modeling, the problem situation usually leads to the appropriate model.
Thus, distribution choice is not an issue. Similarly, a test-offit see Section 3. When such a test is required, however, the classical Pearson Chi-squared statistic is recommended. This statistic, and its use, is well described in many introductory statistics texts. Unfortunately, in many engineering problems the sample size is small.
To obtain visual feedback on how well the estimated model represents the data, simple probability plotting is recommended, when,.. That is, one plots the estimated cdf F qi,; 6 and a suitable sample See, for example, Devore, J. The median plotting position see Section 3. See Example 7. For example, a randomly chosen product specimen is classified, upon inspection, as defective or nondefective. In a destructive performance test, a prototype survives or it fails. See also Section 4. Suppose the engineer contemplates a sequence of n such trials. If the population, from which the sample sequence is randomly chosen, is of finite size N, then it will contain some number M of items that would each produce a trial success s.
The number x of successes s that could turn up in the sample sequence may then be of interest to the engineer. There will be 0 5 M 5 N defectives in that shipment. The above experimental situation is characterized by the following conditions: 1. The sampled population is of finite size N. Variables with pmf 5.
Figure 5. The cdf see Section 1. For example, the probability of finding 3 aces in a fairly dealt hand of 13 cards is since there are 4 aces in a deck of 52 cards. Analogous situations arise in engineering, chiefly in the assessment of produc quality by attributes, when the sampled population is of finite size. When th population size is large, the Hypergeometric distribution is approximated by th Binomial distribution see Chapter 6 and the following section.
Hence, the Binomial distribution can be regarded as a limiting case of the Hypergeometric distribution. The mode value x, is the largest integer k satisfying the relation If the right-hand side of 5. Generate n Uniform random numbers ui on 0,l. That is, Thus, the maximum likelihood estimator of M is the smallest integer M satisfying 5. When 5. The sampling variance of M is obtained, using 5.
The Metalog Distributions
The exact confidence level is the difference between the above cdfs. See Example 5. The power of the test is calculated from If the calculated power is considered insufficient, the sample size n needs to be increased. APPI 5. To monitor the quality of his production, he uses control charts that tell him when he needs to act to maintain acceptable quality levels.
A third use of statistics in quality control is to decide the acceptance of incoming materials that feed the production process. These materials e. The statistical. When quality is measured by an attribute i. The central problem here is to design a sampling plan with desirable characteristics. These characteristics are usually expressed as type I and I1 error probabilities see Sections 3. In this context, the type I error probability a is called the producer's risk: the probability of rejecting a lot of acceptable quality level AQL.
The type I1 error probability B is termed the consumer's risk: the probability of accepting a lot with the worst acceptable i. Since n and x, are integers, an exact solution is rarely obtained. Rather, one chooses a combination n, x, such that a and do not exceed their stipulated values. The characteristics of a chosen sampling plan are displayed by its operating characteristic OC curve, which plots the probability of accepting the lot as a function of the lot quality level M or the fraction defective MI N.
It may then be instructive to also plot the OC curve for the limiting Binomial sampling plan, to see how closely the Binomial solution approximates the exact Hypergeometric solution. For example, suppose that M members of an endangered species are caught, tagged, and released. In a later sample of size n, x tagged specimens are found. Clearly the probability of x is given by the Hypergeometric pmf.
The likelihood of this experiment, in terms of N, is therefore given by 5. However, the moments of 5. A sample of 20 components was inspected and 4 defectives were found. What is the actual confidence level on that interval? Estimate From 5. Confidence interval Expressing the Hypergeometric cdf in terms of factorial functions: X M! Every other component from an improved lot is to be checked for defects. Thus, if 5 or fewer defects are found in the sample, the supplier's claim i s accepted. Power of the test: From 5. The shipment lot size i s In contrast to a finite manufactured lot see Chapter 5 , the population of possible components produced is large.
Even for a well-designed and operated production process, there will be a small proportion of components that do not meet all manufacturing specifications. The production engineer would want to monitor the rate p at which these defectives are produced. To obtain information on p, he would randomly sample n components from the process, have them inspected, and have the number x of defectives counted.
The quantity of interest here is the number X of successes s in n Bernoulli trials. Since the probability of a success is p at any one trial, each arrangement occurs with probability p x l - p n-X. The value of n is usually known from the circumstances of the experiment, whereas p is unknown and needs to be estimated from data.
Discrete random variables X that are distributed according to 6. Figure 6. Several Binomial pmfs. See Example 6. It is one of the oldest distributions studied systematically. Interest in it arose originally in connection with games of chance. The term "Binomial" derives from the fact that the pmf 6.
The most likely mode value of Xis the integer xm satisfying the relation The first and second shape factors see Section 1. The Binomial distribution has a reproductive property. That is, the sum of n independent Binomial variables X, with parameters ni and common p is also Binomial with parameters ni and p: xi - where the symbol denotes "distributed as. Often such systems are studied and optimized by simulating the system's operation. Thus, the Binomial variable X, given p and n, needs to be simulated see Sections 2.
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A Bernoulli trial is simulated quite simply by a single random number u. The trial is a success if u 20 and the value of the p is near 0. The simulated Normal observation see Section The estimator is unbiased and minimum-variance-bound for all n see Section 3. Its standard error is the square root of 6. When the sample size n is large, a simpler, approximate confidence interval on p is obtained from the Normal approximation to the sampling distribution of the maximum likelihood estimator p see Section 3.
Process improvements aim to reduce it, and a statistical test is required see Section 3. That is, when precise measurements of a product characteristic are unnecessary or infeasible, an inspected specimen is simply classified as acceptable or not in terms of a required attribute. For example, a locating pin on a casting needs to be small enough to fit a hole in a mating part. The inspection of a specimen casting would attempt to fit a go-1 no-go gauge to the pin. If the gauge fits, the casting is classified as acceptable. If it does not fit, the casting is classified as a reject. When the population from which specimens are sampled can be considered very large e.
In that case the sample represents, at least approximately, a Bernoulli sequence, and the number X of successes in n trials is a Binomial variable. The control of an in-house production process is the usual application, where the process defect rate p is of central interest. A related application is acceptance sampling by attributes, where the quality level of incoming material is of concern. Here one samples from lots of finite size N.
It is usually impractical to return an inspected item indistinguishably to the lot, so that inspection is without replacement.
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The exact model of the number of successes in n such trials is the Hypergeometric distribution see Chapter 5. To illustrate the approximate nature of the Binomial model for finite lots, consider a lot of items, containing 5 defects. Suppose the first inspected item is defective, at the probability Thus p is not constant and the trials are not independent, so that inspections do not conform to the Bernoulli sequence. However, the Binomial model for the number of defects found in the inspection sample would provide a satisfactory two-decimal approximation for sample sizes less than ten.
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By distribution-free inference is meant a procedure that does not assume a specific distributional form for the measurement variable in question. Such procedures are required when a distributional postulate is not warranted and the data base is deemed too small to fit a distribution to the data. The remainder of this chapter briefly introduces these inferences. The distribution-free estimate of F h is thus given by the success ratio 6.
A distribution-free estimate of t,, is clearly the order statistic t k where equals q or is just less than it. An acceptable range of measurements may be stated as the physical tolerance limits tL,tu within which an inspected product specimen is acceptable. The engineer may then want to know the proportion d of acceptable products he can expect from the production process on average. For example, suppose the diameter of a stubshaft at the bearing seat is of critical importance.
To produce acceptable assemblies, that diameter must be between two physical tolerance limits specified by the design engineer. Stubshafts with diameters outside these limits are classified as defective. Since the estimated model F x will be based on a finite sample of measurements on X, the calculated proportion d will feature statistical uncertainty. Load More Articles.
But a bathtub distribution, as I understand it, is a combination of three different plots—a piecewise plot. I have been a reliability engineer for over three and a half decades. Early, there is at least one infant mortality distribution, with a decreasing failure rate, generally caused by inherent flaws in material, the process, or design capability. In cases where the design itself is capable, a portion of the population will be removed due to failure in this arena. The exponential distribution may overwhelm the infant mortality and wear-out portions of the hazard plot for some time, leading many to utilize only the exponential in reliability demonstration.
This is a risk, because of some inherent properties of the exponential. The first is that not only do infant mortality and wear-out not appear in the exponential distribution, it precludes their existence, instead rolling them into the average failure rate, thereby underestimating both infant mortality and wear-out, and overestimating any constant failure rate. The second is that the mathematics implies that reliability can be determined by either testing one unit for a very long time potentially hundreds of lifetimes , or thousands of units for a very short period potentially only a few minutes worth of stress and state that the product meets reliability goals.
In reality, a reasonable sample size is required to represent some level of variation in the production of the product, and some time that at least includes the period of interest for the evaluation. This article discusses the Weibull distribution and how it is used in the field of reliability engineering.
Learn More About: reliability engineering weilbull failure distribution testing and failure analysis statistics and mathematical analysis. You May Also Like:. Introduction to Reliability in Electronics: Tools and Metrics for Anticipating Device Failure Reliability engineering estimates device lifetimes and failure rates to determine what will fail and when.
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