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 (true mean) (sample mean) (sample variance) (true variance) (sample mean) (sample variance)

Populations Parameters and Sample Statistics Population parameters include its true mean, variance and standard deviation (square root of the variance): 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): Note:

Normally Distributed Population using MATLAB’s command randtool 50 20

Random Sample of 50 49.45 15.72

Another Random Sample of 50 49.86 21.46

Beware of small samples

The Histogram Time record Histogram of digital data Figure 7.3 Figure 7.4 Time record 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: Identify the minimum and maximum values of x and its range where xrange = xmax – xmin. Determine K class intervals (usually use K = 1.15N1/3). Calculate Δx = xrange / K. Determine nj (j = 1 to K) in each Δx interval. Note ∑nj = N. Check that nj > 5 AND Δx ≥ Ux. 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 f3 nj 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?