1_Chapter 4 Probability and Statistics Intro.pdf

# These basic probability definitions are the

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These basic probability definitions are the foundation for advanced statistics in data analysis Random fluctuations in data sets are assumed to follow a predictive distribution of occurrence (pdf) over a certain range of values around the central or mean value Outcome is a true value estimate with an uncertainty interval around it; interval associated with a probability

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Basic Statistics Values
Mean and Standard Deviation Before applying sophisticated analyses, we must always determine the center and the spread of the data These are known as the mean (or average ) and the standard deviation Important note: No statistical analysis should proceed if the mean and standard deviations are not first estimated – These functions are available in all programming languages and business spreadsheets – The mean can only be formulated if the function is “stationary” (the mean does not change in the data set)

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Mean and Standard Deviation Infinite Data Set Continuous Form 1 Again defined differently for continuous functions (look at first) and discrete data sets For a continuous, known function x ( t ) that varies in time or space:... True mean value: Standard deviation: (True variance is ı 2 )

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Mean and Standard Deviation Infinite Data Set Continuous Form 2 Function x ( t ) is an arrangement of an infinite number of x values in time or space. The same values can be restated as x times the probability of occurrence of x : x times a pdf, p ( x ). The integral is transformed from an integral over time or space to an integral over the probability distribution. True mean value: True variance: Standard deviation: ı 2
Common PDFs... Rectangular: random data falls within minimum value a and maximum value b with equal probability of occurrence If b - a = 1, probability that random data point lies within a and a + ½ is 50%, the area under the p( x ) curve between a and a + ½.

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Common PDFs... Rectangular: random data falls within minimum value a and maximum value b with equal probability of occurrence Data below is random, but follows rectangular distribution around y = 1.5.
Common PDFs... Gaussian (or Normal): used for physical properties continuous in time that have variations due to random error Random values clustered around the true mean value; symmetric

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Common PDFs... Gaussian (or Normal): a lot of experimental data recorded has random fluctuations that follow this pdf Æ used for statistics in this course Data below has true mean of 1.5, ı = 0.25, notice clustering and no max/min boundary
Higher Order Moments Although adequate for many types of data, the mean and the standard deviation statistical measures may not be enough The mean is a first order statistic The variance is a second order statistic, but others can be calculated up to 4th order with moment generating function. 3rd order: skew or skewness Æ information about whether the distribution is symmetric 4th order: kurtosis Æ information about peakedness There are examples where these low-order statistics can be misleading, i.e., give a false sense of the “truth”

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Mean and Standard Deviation Infinite Discrete Data Set Defined differently for discrete data sets For an infinite data set of N points : True mean value: True standard deviation: ݔ ே՜ஶ ͳ ± ෍ ݔ ௜ୀଵ ߪ ൌ ே՜ஶ ͳ ± ෍ሺݔ í ݔ ) ௜ୀଵ
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