Lec2 Probability

Lec2 Probability - Physics 257 Lecture 2: Probability (...

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Physics 257 Lecture 2: Probability Prof. Matt Dobbs Physics Department, McGill U. Prof. M. Dobbs, Physics 257 Sept 20, 2007 2 Important Topics from Last Lecture ± Covariance (from last week) ± error propagation with partial derivatives ± error on the mean Δμ = σ / N ± mean, weighted averages 1 ) )( ( = N y y x x s i i i xy y x xy xy s s s r = Prof. M. Dobbs, Physics 257 Sept 20, 2007 3 Histograms ± Area of bins is proportional to the number of occurrences. ± divide full data range in n bins of width Δ x(bin±size±can ±also± be variable) z choose bins carefully. In general σ /2 is a good choice. ¾ too narrow -> few events in bin, fluctuation ¾ too wide -> obscure details Too narrow Too wide Just right. Prof. M. Dobbs, Physics 257 Sept 20, 2007 4 Assignment Feedback the TA would “ be grateful if (the students) would show their uniqueness everywhere except in their handwriting ± For every question (even computer responses), you must show an example calculation. If you are using a spreadsheet, you must print and include the spreadsheet, labeling columns to show the formula that has been used. ± Late assignments (after 1:05pm on lecture day) will lose 25%, like in the lab. z After lecture day the mark will be zero. ± Remember, σ
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Prof. M. Dobbs, Physics 257 Sept 20, 2007 5 Outline ± Questions ? ± Why is s α 1/(N-1) ? ± Probability distribution functions: z Flat distribution z Gaussian Distribution z Poisson Distrtribution ± Famous Experimentalist ± More fun with matlab ± Probability language Prof. M. Dobbs, Physics 257 Sept 20, 2007 6 Review: std deviation, error on mean ± Two researchers measure the period of a pendulum 9 times each. z The researchers have thought about all possible systematic errors and believe these are negligible. ± Researcher1 measures: t 1 , t 2 , …, t 9 z mean = 10.0 seconds, s R1 = 0.6 seconds z The standard deviation s means: _____. z The error on the measured period is: _____________ ± Researcher2 measures: t 1 , t 2 , …, t 9 z mean = 10.2 seconds, s R1 = 0.6 seconds z The error on the measured period is: _____________ ± The two researchers combine their results and find an improved measurement of the period: ____________ Prof. M. Dobbs, Physics 257 Sept 20, 2007 7 Why s α 1/(N-1)? ± Consider a Normal distribution with μ =1, σ =1 ± If we have just one sample of this distribution, x 1 =0.3, then we would calculate <x>=0.3, and s=0 (the later is clearly wrong) ± With 2 samples: 0.3, 1.5 we would calculate <x>=0.9 and z We are seeing a trend here – this equation would consistently UNDERestimate the standard deviation, it is a BIASED estimator. z
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Lec2 Probability - Physics 257 Lecture 2: Probability (...

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