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# Class Note for ASTR 518 at UA

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COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
21
WORDS
KARMA
25 ?

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This 21 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15
The X2 test Assumed parent dist NJ Binned sompie dist Xtv II M5 ji bin no jg gtlt N 7 N Pacj2 2 i1 0139 I i M N Pxj jZI N 39 2 2 X v degrees of Xquot i 1 freedom x2 is the ratio of the observed to the expected difference between sample and assumed parent distributions It constitutes a measure of the GOODNESS OF FIT The x2 probability distribution For 10 degrees of freedom the probability is 060 that XVZ will exceed 0830 for random sampling from the assumed parent A small value for P suggests that the parent may not be correct p112 1 TABLE C x2 distribution Values of the reduced chisquare x lev corresponding to the probability Pxx1 v of exceeding x1 versus the number of degrees of freedom v P 0913 095 090 080 070 060 050 l 0 00393 00158 00642 0143 0275 0455 2 00100 0105 0223 0357 0511 0693 3 00383 067 0195 0335 0475 0 623 0789 4 00742 0107 0266 0412 0549 0688 0839 5 0111 0150 2 0469 0600 0731 0870 6 0145 0139 0273 0367 0391 7 0177 0223 0310 0405 0907 8 0206 0254 0342 0436 0918 9 0232 0281 0369 0463 0927 10 0255 0306 0394 0437 0934 11 02711 0323 0416 0507 0 635 0741 0940 12 0298 0343 0436 0525 0651 0753 01143 0945 13 0316 0367 0453 0542 0664 0764 0856 0949 14 0333 0383 0469 0556 0676 0773 01163 0953 15 0349 0399 0484 0570 0687 07111 0869 0956 We ve already seen x2 J s V 1 ark Pk l i gq 39VU1 CT2HJV f39 xp w E 313311 1 0391 03923 V e l r if 39 Maximizing probability is equivalent to minimi 39 leastsquares for a Gaussian underlying distribution Leastsquares t to a line 0k Ayk yk39a39bxk as before we Wish to minimize 2k Ayk20k2 Uncertainties in the tting coef cients q M 12 Jane H 2i s 3 3M2 i g i 3 i V degrees of freedom Ftest for additional F TABLE C5 F distribution v 1 Values 0 quot corresponding to the probability I FlV v2 0 o of exceeding F with v 1 degrees of freedom versus the larger number of degrees of freedom v2 Degrees o Prohah ty 1P111fexceedingF l I fruedum 1 050 025 010 005 0025 1101 0005 0001 3930 16100 1424111 4050171 5200110 40000011 1 1 551 1350 31150 0850 19800 091111 2 7 2 2 1 202 554 1010 1740 3410 5500 1070 F X1 Xu 1 Ahab 1 4 1111 454 771 1220 2120 11 11 741 5 109 400 001 1000 1030 22110 471 12 1171 Xi lV l 0 5 102 3711 11111 13 70 11100 155 7 500 157 1107 1220 1020 202 11 400 1 54 757 1130 1470 254 9 494 721 1000 1700 231 For degrees of 694 10110 12110 210 11 04110 147 4114 572 9155 1220 107 freedom the probab111ty 12 04114 146 475 655 933 1180 180 15 04711 141 454 020 1108 10110 100 of obtammg F gt 328 for 32 33 10 33 3131 4 934 118 9 19 1 5 2 05 0 random dev1at10ns around 30 0400 1311 417 557 756 1 111 111 411 0403 150 2214 401 542 771 5115 120 a t IS 10 Once agaln 00 0401 115 79 400 529 708 1149 120 120 04511 134 275 302 5 15 0115 11125 114 a large FVa1ue leads to a x 0455 132 271 3114 502 005 71111 1011 Nola For largcr Nine 11139111 pmhub ity P he value nil ix upprmimatci F 251I gt 11 A reallife example linear t 15000 I I I I I I I I I I I I I J I I I 2 2 90II xv 451 forv 10 Pm 10 10000 8 g j o S A Q I 5000 O I I I I I I I I l I I I I 7000 7500 8000 850039 9000 HJD Same data quadratic t F test I I I I l l 1 T l I I I I I F 15 2 2 V 9 xv 093 for v 9 PxV 050 V1500 Cyg Rotational Ephemens 1 T 1987 Mar 1992 Oct E g 0 I 5 E v 5 f LI 39 quotl 3 E3 386 013 x 19739 139 I70 yrgip o gtNt 39 39 05 I I I I I I I I I J H I 39l I I I I I I I I I 5 7000 7500 I 38000 8500 9000 HJD 244100000 The D parameter for the KolmogorOVSmirnov K S test 1 Assumed parent l Observed cumulative probability Computing KS probabilities Onetailed Sample vs Assumed parent m PD gt observed 2 271V 13 2j1V j1 Where A V 012 011WD TWOtailed Sample 1 vs Sample 2 NINZ As 142 quot 7 s pr usng N1 N2 Transformation of probabilities Let q u be continuous functions u with properly normalized prob distributions 4Pq dq 1 etc If u fq then Pudul PqdCI Example Unit vectors on a sphere What is probability distribution for length of the projection of a unit Vector r along a given axis say 2 Example continued We have Pat Pq ldqdul Our example has u gt 2 q gt 6 and r 0036 Then ldqdul ldQdzl 11 zz1 2 1sin6 For random orientations in 3D space P6 sine So Pz sing 39 lsin 1 So probability distribution for length of 3D unit vector projected onto any axis is uniform What is the mean projected length 5 Transformation of Deviates How to transftmn random numbers distributed uniformly over say 0 1 into another distribution 1e px dx dx 0 lt1 x lt1 1 px dx 0 otherwise Want random numbers distributed as Gaussian or exponential or trig function eg pOM mej cb 0ltylt 196 d6 sine d6 u2 lt 6lt viz2 etc Example Deviates distributed as sin 6 We know pa dx pm d6 sin6dt9 Since px 1 dx sin d 80 x fsin d co Normalization oost9 1 so use just one hemisphere 0 We have x6 we want 6 x So nd inverse function Here 6 cos11 Transform random number using this See Press et al for other distributions and subroutines Sampling I oscillation 1x cos27rfIX sampling lll llllll Highest frequency that can be accurately reconstructed is Nyquist critical frequency fc fsZ Sampling II with sampling as 005221Ef5 x 1COS21foD2 The output signal is then 01 t equation Rewriting the cosine product as an angle sum we obtain the wellrkuown quotb 0m 00527rf1m 00527rf1 Hm cos27rf1fm Now note that from mn de nitions above f 2 f Dropping an the 2m and at for the sake of notationa1 brevity we nd 21 goosm 1 2m Input frequency f slightly lower than fc If f 1 A the input signal is modulated at a rate slightly slower than the Nyquist frequency cosf A ms f A cosp A1 0m now an cosf A msf A gmsp A1 mod n 21th recovered I I too fast to track Input frequency f slightly higher than f6 Here we set f j 1 and simplyr illangn the on the s H l 1 39 0L1 il I I ECTUSUQ E EOSFi I A v39 A 2 too fast to trgck alias signal appears as a coherent signal below fc mod n at f1 too fast to track The Sampling Theorem It is not possible to accurately record information on the behavior of a fmetion at frequencies above the critical Nyquist frequency f6 fS2 Corollary Power at frequencies gtfC is aliased into frequencies 92 and appears as a modulation in the sampled signal

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