Lect08 - UCF Physics AST 5765/4762(Advanced Astronomical Data Analysis Fall 2008 Lecture Notes 8 Probability Distributions and Error Analysis 1

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Unformatted text preview: UCF Physics: AST 5765/4762: (Advanced) Astronomical Data Analysis Fall 2008 Lecture Notes: 8. Probability Distributions and Error Analysis 1 Check in: 2:00 — 2:10, 10 min • Did anyone notice errors in Bevington equations? 1 HW problem extra credit for first time each reported. • Bevington Eq 4. on p. 48 (log formula) fill in • Bevington Eq 4.3 on p. 52, Π should be π . • Bevington Eq 6.10 on p. 105, x should be x i in first line and 1 σ 2 i should be x i σ 2 i in last line. 2 Gaussian Distribution: 2:10 — 2:15, 5 min • p G = 1 √ 2 πσ 2 e − 1 2 ( x- x σ ) 2 (1) • Limit of sum of binomial (coin-toss) distribution (and most others) for large N • “Normal” distribution • 1 σ = 65% of measurements closer to mean than this • 2 σ = 95% of measurements closer to mean than this • 3 σ = 99% of measurements closer to mean than this • “Well-behaved” errors follow it • Memorize it! • Not analytically integrable: Find function to calculate integral values. 3 Poisson Distribution: 2:15 — 2:25, 10 min • Timing of random, discrete events in time or space (diagram, don’t confuse x with space!) • Example: Light! • Say there are n events per unit time, on average • Then τ = 1 /n is the average interval between events • In time t we would expect an average of t/τ = tn = N events 1 • The probability of getting x events in time t is P ( x, t ) = ( t/τ ) x x ! e − t/τ = N x x ! e − N (2) • ¯ x = N • σ = √ N • For large N , Poisson approaches a Gaussian with ¯ x = N and σ = √ N • Gaussian, BUT mean and std. dev are locked together! • ¯ x is signal and σ is noise, so S/N = ¯ x σ = N √ N = √ N (3) • Quality of data improves as √ number of counts......
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This note was uploaded on 11/09/2009 for the course AST 4762 taught by Professor Harrington during the Fall '09 term at University of Central Florida.

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Lect08 - UCF Physics AST 5765/4762(Advanced Astronomical Data Analysis Fall 2008 Lecture Notes 8 Probability Distributions and Error Analysis 1

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