Poisson distribution and χ 2 (Chap 11-12)

Transcription

1 Poisson distribution and χ 2 (Chap 11-12) Announcements: Last lecture today! Labs will continue. Homework assignment will be posted tomorrow or Thursday (I will send ) and is due Thursday, February 27 at 4pm in black box. You must work alone and can use only Taylor and class website for resources. I will post clicker scores sometime in the next week on D2L and send . Please make sure your clicker is registered. If you see a 0, please send me your clicker ID (8 digit (hex) number on bar code of back of clicker). 1

2 Clicker question 1 Set frequency to AB You have a large sample of a radioactive element with a long half life (like a billion years). Three students measure the number of decays occurring in 100 seconds in succession. Which statement is most true? A) All three students will find the exact same number B) There will be a clear trend with later students measuring fewer decays than early students. C) There will be a clear trend with early students measuring fewer decays than later students. D) The three students will likely obtain different results but no clear trend. A large sample with a long lifetime means that the decays per second will stay about the same (total population stays ~same) so no trend. Radioactive decays are a random process so the exact number will not always be the same. Governed by Poisson distribution 2

3 Gaussian versus Poisson distribution 3

4 Clicker question 2 Set frequency to AB Assume you have an average count rate of 0.01 counts/second. This means that when you count for 100 seconds you will have, on average, 1 count (but could also have 0, 2, 3, or more). Which statement is most accurate? A) You are most likely to get 1 count after 100 seconds. B) You are most likely to get 0 counts after 100 seconds. C) You are most likely to get 2 counts after 100 seconds. D) You are most likely to get 0.5 counts after 100 seconds. E) Obtaining 0 or 1 count is equally (most) likely. 0 decays: P 1 (0) = e /0! = 1/e = decays: P 1 (1) = e /1! = 1/e = decays: P 1 (2) = e /2! = 1/(2e) = or more decays: P 1 (3+) = P 1 (3) + P 1 (4) + P 1 (5) + + P 1 ( ) Easier way : P 1 (3+) = 1 (P 1 (0) + P 1 (1) + P 1 (2) ) =

5 Poisson è Gaussian 5

6 Radioactive decay example Determine half-life of Radon-220 Determine background rate (no radioactive sample present). Introduce small amount of radioactive material with short halflife (about 1 minute). Count number of decays in 1 second every 30 seconds. Subtract off background rate to get signal rate. Fit the resulting exponential distribution to extract average lifetime. 6

7 Clicker question 3 A) 400 ± 20 counts / second B) 40 ± 20 counts / second C) 40 ± 6 counts / second D) 40 ± 2 counts /second Set frequency to AB The background rate (number of counts in 1 second) was measured 10 times (see table). What is the best estimate of the background rate? Can get the rate by taking the average of the ten trials: 400/10 = 40 counts/sec. This is the same as counting for 10 seconds and dividing by 10. Trial Counts The uncertainty in the number of counts is 9 33 counts so 400 ± 400 = 400 ± 20 counts 400 ± The rate is: = 40 ± 2 counts/sec 10 Total 400 Uncertainty is square root of counts, not count rate. 7

8 Signal data Small sample of radioactive material is added and counts are taken for 1 second every 30 seconds as the sample decays. Time Counts 0 s s s s s s s s s s s s s s s 51 Physics 2150 Spring

9 Signal data Small sample of radioactive material is added and counts are taken for 1 second every 30 seconds as the sample decays. Uncertainties on each count are: σ s = S Time Counts 0 s 2007 ± s 1464 ± s 973 ± s 698 ± s 526 ± s 353 ± s 285 ± s 217 ± s 150 ± s 122 ± s 112 ± s 86 ± s 78 ± s 53 ± s 51 ± 7 Physics 2150 Spring

10 Signal data Small sample of radioactive material is added and counts are taken for 1 second every 30 seconds as the sample decays. Uncertainties on each count are: σ s = S Each count contains signal and background so subtract the background rate of 40 counts/sec (uncertainty is σ b = 2 counts/sec). Time Counts Corrected 0 s 2007 ± s 1464 ± s 973 ± s 698 ± s 526 ± s 353 ± s 285 ± s 217 ± s 150 ± s 122 ± s 112 ± s 86 ± s 78 ± s 53 ± s 51 ± 7 11 Physics 2150 Spring

11 Signal data Small sample of radioactive material is added and counts are taken for 1 second every 30 seconds as the sample decays. Uncertainties on each count are: σ s = S Each count contains signal and background so subtract the background rate of 40 counts/sec (uncertainty is σ b = 2 counts/sec). Uncertainty on corrected rate comes from error propagation: σ c = σ s 2 +σ b 2 Uncertainties are well measured and not all the same. So should use weighted fit. Time Counts Corrected 0 s 2007 ± ± s 1464 ± ± s 973 ± ± s 698 ± ± s 526 ± ± s 353 ± ± s 285 ± ± s 217 ± ± s 150 ± ± s 122 ± ± s 112 ± ± s 86 ± 9 46 ± s 78 ± 9 38 ± s 53 ± 7 13 ± s 51 ± 7 11 ± 7 Physics 2150 Spring

12 Clicker question 4 Fitting to radioactive decay data One can fit the data directly to an exponential function. Or, one can take the logarithm and fit to a linear function. Logarithm of N = N 0 e λt is ln(n) = ln(n 0 ) λt, which is the form of a line y = A + Bx where y=ln(n) and x = t. What is the uncertainty on y if the uncertainty on N is σ N? A) σ y = ln(σ N ) B) σ y = σ N /N C) σ y = N D) σ y = y E) σ y = σ N / N δq = For q(x,y): " q $ # x δx % ' & Set frequency to AB 2 Last topic: How do we know the fit is good? seconds 2 " + $ q # y δy % ' & 2 Counts / second ln(counts) / second seconds

13 Goodness-of-fit (Chapter 12) χ 2 test is a particular type of goodness-of-fit test For N measurements (O 1,O 2,,O N ), " χ 2 = N i=1 O i E i % $ ' # & where E i is the expected value and σ i is the uncertainty on O i. To use, need to know σ i. Could be O i for counting experiment, an estimate of the measurement uncertainty, or the standard deviation if each O i is the average of several measurements. Also need the degrees-of-freedom (dof), generally the number of data points minus the number of free parameters in the fit. Reduced χ 2 is χ 2 /dof and should be ~1. Can convert to probability using Appendix D or in Mathematica: Compatibility prob = 1 - CDF[ChiSquareDistribution[dof], chi2 Normally we want this to be >5% to consider the fit to be adequate (although >1% might be OK in some cases). Covered in more depth in Chapter σ i 2

14 χ 2 test and least-squares fitting Two options for fitting and χ 2 tests: 1. Assume uncertainties are unknown and do an unweighted fit Described in lectures 4 and 5 and in Chapter 8 of Taylor This returns function parameters (usually slope and intercept) with uncertainties on parameters and uncertainties on σ y (from spread in measurements about function). Cannot do χ 2 test to check goodness-of-fit 2. Use known uncertainties and do a weighted fit Described in Problems 8.9 and 8.19 of Taylor This returns function parameters (usually slope and intercept) with uncertainties on parameters; σ y is input from data. Can check goodness-of-fit using χ 2 test. 14

15 Revisit fit to radioactive decay data The fit to data with background subtraction gives χ 2 =10.62 with 13 degrees of freedom. This gives a probability of 64%, which is good. The fit to data without background subtraction gives χ 2 =84.25 with 13 degrees of freedom. This gives a probability of , which is terrible. The χ 2 can be used to tell if a fit is a good match to the data (if the uncertainties are correct). If the fit is bad, probably need a different function. ln(counts) / second ln(counts) / second Fit with background subtraction / seconds / 13 Fit without 3 background subtraction seconds 15

16 Summary of what you should know Error propagation for an arbitrary equation like what is σ q if q(a,b,c,d) = a cos(3b) e 2c / d 3, when σ a, σ b, σ c, σ d are known Understand differences between statistical and systematic uncertainties. Calculate mean, standard deviation, and uncertainty on the mean, and know what they represent. Determine if two measurements are compatible, including the probability of getting a value at least as far away from the true value as your measured value. Determine the weighted average and its uncertainty. Perform least-squares fit to data and extract slope and intercept, with uncertainties. Both weighted and unweighted. Determine if two variables are correlated. Know what a counting experiment is and that the uncertainty on the counts is the square root of the number of counts. Evaluate the goodness-of-fit using the χ 2 test. 16

17 Last comments for lecture The absolutely necessary ingredients for any lab writeup is: Clear explanation of how you got the results A calculation of the statistical and systematic uncertainties A comparison with the known value, including the probability that they are compatible. The homework will be posted some time tomorrow or Thursday (I will send ). It will be due by Thursday, February 27 at 4pm in the black box. You must work on it alone and you are only allowed to use Taylor and the material on the class website (including the lecture notes). 17