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CMU CS 15410 - Investigation of Combined Positive and Negative Muon Decay in a Scintillator Appendices

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Investigation of Combined Positive and Negative Muon Decay in a ScintillatorAppendicesBrian Lim and D. RubenCornell University, Ithaca, NY 14853, USA(received 7 November 2005)IAppendix A: Weighted Least Squares FittingDeveloped from Taylor's discourse16, the following derives the determination of the slope of a linear fit to data where each dependent variable, yi, have different errors,i. Assuming no error in the independent variable, xi, the probability of obtaining yi isProbA , B yi∝1ie− yi− A− Bxi2/2 i2The probability of obtaining the set of yi's is thenProbA, B y1,... , yN=ProbA , B y1⋯ProbA , B yN∝11e− y1− A− Bx12/ 212⋯1Ne−yN− A− BxN2/ 2 N2=11⋯Ne−2/ 2where2=∑i =1N yi−A− Bxi2i2. The best fit would have2minimized by setting its partial derivatives with respect to A and B equal to zero and solving,∂2∂ A=−2∑i=1N1i2 yi−A−Bxi=0∂2∂ B=−2∑i=1Nxii2 yi−A−Bxi=0wherewi=1i2to giveA=∑wx2∑wy−∑wx∑wxyB=∑w∑wxy−∑wx∑wy=∑[Swwixi−Swxwi]yi(A.1)(A.2)where=∑w∑wx2−∑wx2, Sw=∑w,Swx=∑wx.Using the method of quadrature for error propagation, the error in B can be found.II B=∑∂ B∂ yi yi2=∑∂ B∂ yi21wi=∑Swwixi−Swxwi21wi=12∑Sw2wixi2Swx2wi−2 SwSwxwixi=12Sw2SwxxSwx2Sw−2 SwSwx2=Sw2SwSwxx−Swx2=∑w2=∑w(A.3)IIIAppendix B: Data FilesFor data collected after 7th October 2005, the PMTs were set at 1.005V, at the middle of the voltage plateau of the scintillator/PMT setup for most accurate counting of muon detection.oct12.txt 2 0 0 0 0 28 1678 1870 1743 1706 1579 1577 1489 1341 1356 1278 1205 1198 1083 1057 963 934 941 865 845 823 777 744 665 614 635 598 591 585 548 512 506 436 428 392 433 410 351 328 329 315 327 328 293 287 240 243 222 226 234 224 221 209 175 187 171 175 182 154 168 115 135 119 117 125 119 123 93 100 88 108 103 77 89 95 85 80 71 71 72 74 56 67 69 66 64 71 63 42 47 35 57 52 51 37 51 54 39 30 39 48 42 36 39 41 29 38 18 24 36 28 29 32 29 31 36 41 43 33 36 35 26 31 34 35 31 23 32 38 29 25 25 25 18 29 22 23 20 25 24 19 29 25 32 17 23 25 28 23 20 26 35 20 25 32 29 24 31 19 29 24 29 24 24 27 18 31 23 21 23 30 27 25 27 31 22 32 24 20 30 25 19 30 31 23 30 27 24 18 8 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 3 0 0 0 1 0 0 423399 oct14.txt 2 0 0 0 0 11 607 670 589 594 594 592 527 551 494 440 415 406 382 365 379 316 308 312 271 287 264 244 257 268 213 205 200 202 168 180 162 161 162 145 137 138 116 150 109 133 109 104 107 114 87 101 89 92 79 83 58 75 68 63 53 65 60 54 41 51 59 52 51 36 39 51 43 35 40 36 37 19 39 32 21 21 27 30 21 29 23 20 26 19 19 23 16 25 18 18 22 16 19 15 19 13 18 13 12 18 19 12 9 18 22 12 13 13 12 14 5 7 11 19 13 17 20 12 15 16 11 4 11 8 11 7 16 11 7 11 9 10 11 3 10 10 5 6 12 15 5 7 12 5 14 7 14 9 6 10 16 10 13 10 7 7 3 13 9 7 8 9 11 6 5 8 9 10 12 8 13 10 11 13 7 9 8 6 7 6 6 11 7 9 10 8 4 5 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 153504IVoct26.txt 11 0 0 0 0 89 4278 4842 4393 4298 4148 3942 3818 3768 3340 3353 3039 2891 2891 2694 2629 2491 2353 2249 2192 2033 1946 1873 1819 1704 1610 1575 1491 1328 1378 1312 1231 1157 1077 1089 1017 993 990 839 869 791 793 737 723 658 649 640 621 620 583 546 539 529 483 440 475 394 405 384 358 339 356 348 337 303 277 283 270 259 248 264 227 203 210 219 195 212 190 148 176 183 182 161 184 153 141 154 132 141 141 127 125 118 120 137 109 102 113 113 130 105 107 88 112 91 96 97 77 103 91 90 93 81 106 85 85 74 71 77 76 76 87 67 82 68 87 79 60 73 74 72 73 69 70 76 77 73 59 63 64 68 67 82 70 66 70 64 63 64 65 64 74 57 63 72 70 62 68 55 64 63 57 61 47 65 53 68 59 55 70 74 63 62 76 74 69 48 56 66 66 65 55 59 64 53 72 56 66 43 12 0 0 1 0 0 0 0 0 0 0 2 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 0 2 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 2 0 1 0 1 1 1 1 0 0 1 0 0 1 2 0 1036965nov01.txt 9 0 0 0 0 62 2260 2401 2217 2278 2110 2040 1881


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CMU CS 15410 - Investigation of Combined Positive and Negative Muon Decay in a Scintillator Appendices

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