Physics and Engineering of Radiation Detection

536 Chapter 9. Essential Statistics for Data Analysis quotinghand hmight not be sufficient and one should present the distributi ...

9.3. Probability 537 Hence maximum of ln(L)atp∗is r p∗ + N−r 1 −p∗ =0 ⇒p∗ = r N . (9.3.28) Now in order to evaluate the error in ...

538 Chapter 9. Essential Statistics for Data Analysis x 0 1020304050 f 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 μ=5.5 μ=10 ...

9.3. Probability 539 be the square root of the measured quantity. For example, if we count the number of γ-ray photons coming fr ...

540 Chapter 9. Essential Statistics for Data Analysis x 0 5 10 15 20 25 30 35 40 45 f 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 ...

9.3. Probability 541 likelihood solution will become μ∗ = ∑N ∑i=1xi/σ N i=1(1/σ) = 1 N ∑N i=1 xi, (9.3.43) which is nothing but ...

542 Chapter 9. Essential Statistics for Data Analysis 9.3.41 again with respect toμ∗and substituting the result in the above exp ...

9.3. Probability 543 Suppose we havemindependent normally distributed random variablesuihaving theoretical meansμiand variancesσ ...

544 Chapter 9. Essential Statistics for Data Analysis distribution, let us first write z = ∑n i=1 x^2 i and t = x √ z/n , fornin ...

9.3. Probability 545 Now that we have learned all the basics of maximum likelihood methodology, we are ready to use it in practi ...

546 Chapter 9. Essential Statistics for Data Analysis Withf=kx,weget ( ∂f ∂k ) 2 = x^2 ⇒ 1 f ( ∂f ∂k ) 2 = x k ⇒ ∫ 1 0 1 f ( ∂f ...

9.4. Confidence Intervals 547 If the functionL(x) is normalized, then the denominator becomes 1 and the probability is simply gi ...

548 Chapter 9. Essential Statistics for Data Analysis with respect toσ,welookatsometypicalvalues. P(μ−σ<x<μ+σ)=0. 6827 P(μ ...

9.5. Measurement Uncertainty 549 having better accuracy or, in case of a gas filled detector, improve on its accuracy by using a ...

550 Chapter 9. Essential Statistics for Data Analysis These general relations can be used to derive formulae for specific functi ...

9.6. Confidence Tests 551 9.5.D PresentationofResults Now that we know how to calculate errors associated with parameters by usi ...

552 Chapter 9. Essential Statistics for Data Analysis cumulative distribution function p ≡ 1 −P(to)=1− ∫to −∞ g(t|h)dt or p = ∫∞ ...

9.6. Confidence Tests 553 7.Compareχ^2 /νwithχ^2 ν,α/ν. Let us now see what we can infer from this comparison.
Case-1,χ^2 /νχ^ ...

554 Chapter 9. Essential Statistics for Data Analysis generally given for different degrees of freedom and levels of significanc ...

9.7. Regression 555 The standard deviation of the mean is given by σ 12 = [ σ^21 N 1 + σ^22 N 2 ] 1 / 2 = [ 17. 612 5 + 20. 592 ...