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Probability “When you deal in large numbers, probabilities are the same as certainties. I wouldn’t bet my life on the toss of a single coin, but I would, with great confidence, bet on heads appearing between 49 % and 51 % of the throws of a coin if the number of tosses was 1 billion.” Brian Silver, 1998, The Ascent of Science, Oxford University Press. Simple Probability Problem • Imagine I randomly choose 2 people from this class. What is the probability that both are in the same laboratory section? • Assume: 99 students, all present; 9 lab sections, all equally populated 11 students per lab section • Choose 1st student (note this choice can’t be wrong) • Now there are 98 students left and 10 that are in the same section as the first… • Thus the answer is 10/98 = 10.2% Sample vs Population x 2 (true mean) (true variance) (sample mean) x (sample variance) 2 Sx Populations Parameters and Sample Statistics • Population parameters include its true mean, variance and standard deviation (square root of the variance): x lim N 1 N 2 lim N N x 1 N i 1 i N 2 ( x x ) i i 1 • Sample statistics with statistical inference can be used to estimate their corresponding population parameters to within an uncertainty. Populations Parameters and Sample Statistics • A sample is a finite-member representation of an ‘infinite’-member population. • Sample statistics include its sample mean, variance and standard deviation (square root of the variance): 1 x N N x i 1 i N 1 2 S x2 ( x x ) i N 1 i 1 Note: 1 1 as N N 1 N Normally Distributed Population using MATLAB’s command randtool Distribution Samples 4500 4000 3500 Counts x x 3000 x x 50 20 2500 2000 1500 1000 500 0 -100 -50 0 50 Values 100 150 200 Random Sample of 50 Distribution Samples 18 16 x 49.45 S x 15.72 14 Counts 12 10 8 6 4 2 0 -100 -50 0 50 Values 100 150 200 Another Random Sample of 50 Distribution Samples 25 x 49.86 S x 21.46 20 Counts 15 x x Sx 10 5 0 -100 -50 0 50 Values 100 150 200 Beware of small samples The Histogram Figure 7.3 Time record Figure 7.4 Histogram of digital data analog, discrete, and digital signals 10 digital values: 1.5, 1.0, 2.5, 4.0, 3.5, 2.0, 2.5, 3.0, 2.5 and 0.5 V resorted in order: 0.5, 1.0, 1.5, 2.0, 2.5, 2.5, 2.5, 3.0, 3.5, 4.0 V N = 9 occurrences; j = 8 cells; nj = occurrences in j-th cell n5 = 3 The histogram is a plot of nj (ordinate) versus magnitude (abscissa). Proper Choice of Δx High K small Δx The choice of Δx is critical to the interpretation of the histogram. theoretical values data (5000 randomly drawn values) Figure 7.5 Histogram Construction Rules To construct equal-width histograms: 1. Identify the minimum and maximum values of x and its range where xrange = xmax – xmin. 2. Determine K class intervals (usually use K = 1.15N1/3). 3. Calculate Δx = xrange / K. 4. Determine nj (j = 1 to K) in each Δx interval. Note ∑nj = N. 5. Check that nj > 5 AND Δx ≥ Ux. 6. Plot nj versus xmj,where xmj is the midpoint value of each interval. Frequency Distribution The frequency distribution is a plot of nj /N versus magnitude. It is very similar to the histogram. n3 nj f3 fj = nj/N Figure 7.7 Histograms and Frequency Distributions in LabVIEW ‘digital’ case ‘continuous’ case • odds to get something far from mean? • effect of noise form, e.g. uniform noise?