1_Chapter 4 Probability and Statistics Intro.pdf

Mean and standard deviation finite discrete data set

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Mean and Standard Deviation Finite Discrete Data Set Defined differently for discrete data sets For a finite-sized data set of N points: Sample mean value: Sample standard deviation:
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What we want with a data set… Suppose the true value is x ƍ (and we don’t know it). It is estimated from the data set with the sample mean, the uncertainty interval, and the probability level at which true value lies within the interval defined around the mean “We estimate, with 95% probability (confidence interval), that the true value is in the range of 123 ± 2 psi” ݔ ൌ ݔ ҧ േ ݑ ሺܲΨሻ
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What we want with a data set… Confidence interval values u x (P%) can be determined with the Normal distribution and/or a modified Normal distribution known as the Student’s t distribution. Need to know more about the Normal distribution.
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Need to Know Independence or dependence of random variables Is random variable Continuous or Discrete Mean, Variance, Standard Deviation P( A ) Probability that A will occur Summation of all Probabilities for the all possible outcomes of a single random variable is 1. p( x ) probability distribution function for x The probability that an outcome between x 1 and x 2 will occur is the area under probability density function pdf for x between x 1 and x 2
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Summary
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Random variable characteristics A random variable x may take on many values (Outcomes) The trial of a random variable x produces an outcome. The outcome of x may differ for each trial of x . The Possible number of Outcomes for x can be Fixed or Infinite depending on the situation. The random variable is independent if its outcome does not depend on the outcome of any other random variables. Otherwise it is dependent. The summation of all possible outcomes of a single trial of an independent random variable x must be 1
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Describe random variable behavior The probability that x will produce outcome A is written as P(A) Multiple outcomes of a single random variable or multiple random variables can be described by multiplication or addition of the probabilities of a given outcome. Each outcome has a probability of occurring and a historical frequency of occurrence. Possible future or past recorded outcomes can be presented as a histogram (discrete)
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Describe random variable behavior For a Random Variable x with infinite number of possible outcomes. A finite set of outcomes is described by a mean and standard deviation Frequency of occurrences in a range is described by the area under a pdf (probability density function) curve associated with the range The total area under any pdf is 1. A finite set of discrete outcomes will not necessarily contain the outcomes predicted by the pdf. The predictions for future behavior of x must be adjusted by the number of samples (trials) in the set.
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