Class Note for ASTR 518 at UA
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This 14 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 13 views.
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Date Created: 02/06/15
Defining 7 by maximum likelihood Consider a set of N observations of quantity x assumed to be drawn from a Gaussian parent Probability of obtaining any given xk is 1 iz uz Palm Wm Probability of obtaining actual set of N measurements is joint probability N Pltm1m2gt Hme 1 Gaussian maximum likelihood Let s define a sample mean f in a way that the probability of obtaining the observed set of measurements is a maximum This will be our best estimate of the parent mean of the Gaussian u Write 1 P T171727quot39 OW y exp i I 122 1 Gaussian maximum likelihood We maximize the joint probability by differentiating wrt Z and setting to zero Since dez ez dz this means the delivative of the exponent equals zero 0Pzl r2 7 7 19 are 0 7 07 0392 0 zkia39V 7 3 lt72 We conclude for an assumed Gaussian parent 39 The standard de nition of the sample mean 339 is also the best estimate for the mean of the parent distribution th The standard de nition of the mean minimizes the quadratic sum of the deviations about that mean The mean f for unequal weights N 1 13 A 17 393 1 n1 1 7 1 r quot 111 513 n T1 T3 W 2311 e L 2 v 3 111 Uk kl quot7 3395quot 2 3PIT1I39 HUI 03923 3 Z ark 17 e II 739 r if 6 aquot kl 213 i 7 2 Us ZquotT Tk 339 1 The variance 03920 or m E IMF Once the sum in the denominator is performed it can be extracted from the outm sum 1 2 flnail 1 or for equal weights 023 02 N Sample variance vs Parent variance 0392 VS 52 c7 characterizes width of parent distribution around parent mean u s characterizes width of sample distribution around sample mean Y For proper choice of parent as N gt so Ya u s gt a39 We now return to that N l in the de nition of the sample variance Rewrite the de nition of the parent variance N aoo hm Ztxk a m or N mo N 1 72 11m Nxkp2 1 dispersion of values difference between around sample mean sample mean and parent expand mean 02 ljm i2 3f p f 02 N gtoo N N N 0 0 for limit N gt 00 For nite N mean square deviation of 7 from y or 0 Since we are trying to approximate aby s for nite N we have 2 if 025 substituting 026 0392 N s2 N for finite N N N 821 Z T 2 TV Statistics summary to date 8 Equal Weights U71 r qual Weights 1 N k 7 1 Nlc1 Example Rcmm to the case of a CCD A man realistic equation for the digital signal volts or DN you measure is f G 39n b where G is the elactmnic gain in DNlplwtoelectmn n is the number of photaclectmns b is an cleanunit bias in DN Thmi a U 32 m 021 01 is called the read noise but it is generally expressed in equivalent mm photoeleauons as n 65 16 and clay G2 m Example continued t Now a real measurement mquires that the bias be subtracted If We have an accurate measmc for Efmm say nonlight sensitive pixels then fGm b 17 a3f32nn2 SMfiafnnnfm Which highlights the importance of minimizing read noise Note from above we can write for a constant light scum a2f 1462 my Determining CCD characteristics Plot 02f vsf for a range of light levels Slope 2 G DNe yintercept is square ofread noise in DNZ 021 G2 nf so called photon transfer curve lG 59 e DN n 149 e 0 mo 20v 300 Am ADU abuve blas CCD noise sources Notes Modern CCDs achieve read noise of 23 at slow readout rates gt10 itspixel Noise increases as sample time decreases due to lf character of electronic noise Several other sources of CCD noise Flat eld uncertainty Cosmic rays Dark mums Gain uncertainty Charge transfer inef ciency Saturation See Jancsick J Scienti c ChargeCoupled Devices Howell S Handbook of CCD Astronomy
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