Counting Statistics and Error Prediction Chapter3 Counting Statistics and Error Prediction GK-NT-Lec-4 Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. 1

© 2014 John Wiley & Sons, Inc. All rights reserved. Statistical Models 70 I. Characterization of Data 66 II. Statistical Models 70 III. Applications of Statistical Models 79 IV. Error Propagation 85 v. Optimization of Counting Experiments 92 VI. Limits of Detectability 94 VII. Distribution of Time Intervals © 2014 John Wiley & Sons, Inc. All rights reserved.

2.8………Statistical Models 70-1-NT © 2014 John Wiley & Sons, Inc. All rights reserved.

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2.8……Examp,,…Statistical Models 70-4-NT © 2014 John Wiley & Sons, Inc. All rights reserved.

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2.10…….II. Statistical Models-15-NT 2.10.2 Importance of the Gaussian Distribution for Radiation Measurements The normal distribution is the most important distribution for applications in measurements. almost any type of measurement that has been taken many times, the frequency with which individual results occur forms, to a very good approximation, a Gaussian distribution centered around the average value of the results. The greater the number of trials, the better their representation by a Gaussian. even if the original population of the results under study does not follow a normal distribution, their average does. That is, if a series of measurements of the variable is repeated M times, the average values , follow a normal distribution even though the xi's may not. This result is known as the central limit theorem and holds for any random sample of variables with finite standard deviation. no distribution of experimental data can be exactly Gaussian, since the Gaussian extends from -  to +. both the binomial (Fig. 2.2) and the Poisson (Fig. 2.3) distributions ressemble a Gaussian under certain conditions. The results of radiation measurements are, in most cases, expressed as the number of counts recorded in a scaler. These counts indicate that particles have interacted with a detector and produced a pulse that has been recorded. The particles, in turn, have been produced either by the decay of a radioisotope or as a result of a nuclear reaction, the emission of the particle is statistical in nature and follows the Poisson distribution. However, as indicated in Sec. 2.9, if the average of the number of counts involved is more than about 20, the Poisson approaches the Gaussian distribution. For this reason, the individual results of such radiation measurements are treated as members of a normal distribution. © 2014 John Wiley & Sons, Inc. All rights reserved.

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2.11 THE LORENTZIAN DISTRIBUTION-NT-18 © 2014 John Wiley & Sons, Inc. All rights reserved.

2.12 THE STANDARD, PROBABLE, AND OTHER ERRORS-NT-19 © 2014 John Wiley & Sons, Inc. All rights reserved.

2.12 THE STANDARD, PROBABLE, AND OTHER ERRORS-NT-20 © 2014 John Wiley & Sons, Inc. All rights reserved.

2.13 THE ARITHMETIC MEAN AND ITS STANDARD ERROR NT-21 © 2014 John Wiley & Sons, Inc. All rights reserved.

2.13 THE ARITHMETIC MEAN AND ITS STANDARD ERROR NT-22 © 2014 John Wiley & Sons, Inc. All rights reserved.

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2.14 CONFIDENCE LIMITS-NT-25 © 2014 John Wiley & Sons, Inc. All rights reserved.

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2.14 CONFIDENCE LIMITS-NT-29 The use of the concept of confidence limits is widespread in industry. Example, let us assume that x is the thickness of the cladding of a reactor fuel rod. The average (nominal) thickness is xn. The reactor designer would like to be certain that a certain fraction of fuel rods will always have thickness within prescribed limits. Let us say that the designer desires a confidence limit of 99.87 percent. This means that no more than 13 rods out of 10,000 will be expected to have cladding thickness exceeding the nominal value by more than three standard deviations (Table 2.2). © 2014 John Wiley & Sons, Inc. All rights reserved.

III. APPLICATIONS OF STATISTICAL MODELS-GK-30 © 2014 John Wiley & Sons, Inc. All rights reserved.

III. APPLICATIONS OF STATISTICAL MODELS-GK-31 © 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved. Figure 3.9 shows the chain of events that characterizes this application of counting statistics. Properties of the experimental data are confined to the left half of the figure, whereas on the right side are listed properties of an appropriate statistical model. We start in the upper-left corner with the collection of N independent measurements of the same physical quantity. These might be, for example, successive 1-minute counts from a detector. Using the methods outlined in Section I, we can characterize the data in several ways. The data distribution function F(x) as defined in Eq. (3.3) can be compiled. From this distribution, the mean value and the sample variance s2 can be computed by the formulas given in Eqs. (3.5) and (3.9). Recall that the mean value gives the value about which the distribution is centered, whereas the sample variance s2 is a quantitative measure of the amount of fluctuation present in the collection of data. We now are faced with the task of matching these experimental data with an appropriate statistical model. Almost universally we will want to match to either a Poisson or Gaussian distribution (depending on how large the mean value is), either of which is fully specified by its own mean value . What should we choose for ? We would be rather foolish if we chose any value other than , which is our only estimate of the mean value for the distribution from which the data have been drawn. Setting = then provides the bridge from left to right in the figure so that we now have a fully specified statistical model. © 2014 John Wiley & Sons, Inc. All rights reserved.

III. APPLICATIONS OF STATISTICAL MODELS-GK-32 © 2014 John Wiley & Sons, Inc. All rights reserved.

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III. Application A-APPLICATIONS OF STATISTICAL MODELS-GK-37 © 2014 John Wiley & Sons, Inc. All rights reserved.

III. Application B-APPLICATIONS OF STATISTICAL MODELS-GK-38 Application B: Estimation of the Precision of a Single Measurement A more valuable application of counting statistics applies to the case in which we have only a single measurement of a particular quantity and wish to associate a given degree of uncertainty with that measurement. To state the objective in another way, we would like to have some estimate of the sample variance to be expected if we were to repeat the measurement many times. The square root of the sample variance should be a measure of the typical deviation of any one measurement from the true mean value and thus will serve as a single index of the degree of precision one should associate with a typical measurement from that set. Because we have only a single measurement, however, the sample variance cannot be calculated directly but must be estimated by analogy with an appropriate statistical model. The process is illustrated in Fig. 3.12. Again, the left half of the figure deals only with experimental data, whereas the right half deals only with the statistical model. We start in the upper-left comer with a single measurement, x. If we make the assumption that the measurement has been drawn from a population whose theoretical distribution function is predicted by either a Poisson or Gaussian distribution, then we must match an appropriate theoretical distribution to the available data. For either model we must start with a value for the mean x of the distribution. Because the value of our single measurement x is the only information we have about the theoretical distribution from which it has been drawn, we have no real choice other than to assume that the mean of the distribution is equal to the single measurement, or . Having now obtained an assumed value for x, the entire predicted probability distribution function P(x) is defined for all values of . We can also immediately find a value for the predicted variance cr2 of that distribution. We can then use the association that, if the data are drawn from the same distribution, an estimate of the sample variance s2 of a collection of such data should be given by 2• Through this process we have therefore obtained an estimate for the sample variance of a repeated set of measurements that do not exist but that represent the expected results if the single measurement were to be repeated many times. The conclusion we reach can then be stated as follows: © 2014 John Wiley & Sons, Inc. All rights reserved.

III. Application B-APPLICATIONS OF STATISTICAL MODELS-GK-38 © 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved. The expected s2 ,...., cr2 of the statistical model from which we think the sample variance measurement x is drawn = x provided the model is either Poisson or Gaussian ,...., x because x is our only measurement on which to base an estimate of x © 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved.