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## Counting Statistics and Error Prediction

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**Counting Statistics and Error Prediction**Poisson Distribution (p << 1) Success ≡ Birthday today. p = 1/365. n = 1000. • Low cross section. • Weak resonance. • Short measurement (compared to t1/2). Appendix C HW 25 We need to know only the product. Asymmetric Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Gaussian (Normal) Distribution (p << 1, > 20) Success ≡ Birthday today. p = 1/365. n = 10000. HW 26 Symmetric and slowly varying • Can be expressed as a function of . • Can be expressed in a continuous form. Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**• Calculate the percentage of the samples that will deviate from the mean by less than: • one . • two . • etc … HW 27 Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Total area under the curve above the baseline Baseline offset HW 28 2, approximately 0.849 the width of the peak at half height This model describes a bell-shaped curve like the normal (Gaussian) probability distribution function. The center x0 represents the "mean", while ½ w is the standard deviation. What is FWHM? Resolution? Peak centroid? Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Applications1- Match experiment to model • Assume a specific distribution (Poisson, Gaussian). • Set distribution mean to be equal to experimental mean. • Compare variance to determine if distribution is valid for actual data set (Chi-squared test). and not because we set Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Back to our example • We can’t use Gaussian model for this data set. Why? • Qualitative comparison. • Is 2 close to s2? • Close!? Less fluctuation than predicted! • But quantitatively? • Chi-squared test. HW 29 Only to guide the eye! Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Chi-squared By definition: Thus: or The degree to which c2 differs from (N-1) is a measure of the departure of the data from predictions of the distribution. Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**smaller <= Fluctuation => larger Perfect fit 15.891 (interpolation) or http://www.stat.tamu.edu/~west/applets/chisqdemo.html either gives = 0.6646 Conclusion: no abnormal fluctuation. Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Single measurement S2 = 2 ≈ x 68% probability that this interval includes the true average value. What if we want 99%..? Fractional standard deviation Need 1%? Count 10000. Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**A series of “single” measurements. Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**• What about derived quantities? (Error propagation). Net counts = Gross counts – Background Compare to addition instead of subtraction. (Count, stop, count). Derived Measured Gross counts = 1000 Background counts = 400 Net counts = 600 37 (not 600 24) Count Rate = ? ? Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Mean value of multiple independent counts. • Assume we record N repeated counts from a single source for equal counting times: • For Poisson or Gaussian distributions: • So that Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Standard error of the mean • To improve statistical precision of a given measurement by a factor of two requires four times the initial counting time. Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**Optimizing counting time. How to divide the limited available beam time? S = (net) source count rate. B = background count rate. TS+B = time to count source + background. TB = time to count background. To minimize s : High signal-to-background ratio Low-level radioactivity Weak resonance Very strong background HW 30 Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).**Counting Statistics and Error Prediction**• Background measurement? • Without the “source”. • Should include all sources except the “source”. • Accelerator applications: background with beam. • Minimum detectable amount. • False positives and false negatives. • Rest of Chapter 3. Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).