APME_2 - Applied Probability Methods for Engineers Slide...

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Applied Probability Methods for Engineers Slide Set 2
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Chapter 6 Descriptive Statistics
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Data and Samples When we want information about a phenomenon or characteristic, often cannot have complete information We collect as much data as is practical from the entire set of possibilities, or population Given a random variable X, we obtain a sample of observations of the phenomenon, x 1 , x 2 , …, x n We analyze sample data and use this analysis to make inferences about the population To ensure as accurate a representation of the population as possible, we need to ensure that we take a random sample
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Data Presentation Bar chart Machine Breakdowns
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Data Presentation Pareto chart
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Histograms Used to show frequency of occurrences (or distribution) of numerical data Similar to a probability mass function (pmf) A pmf is a histogram for a population (or entire sample space)
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Histograms Positive skewness (right skewed) Negative skewness (left skewed)
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Histograms Bimodal distribution Outliers
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Sample Statistics Sample mean of a data set is arithmetic average Sample median is the “middle” value (if n even, average value n/2 and n/2 Sample mode is the most commonly occurring value Sample variance 1 / n i i x x n = = ( 29 ( 29 2 2 2 2 2 1 1 2 1 1 1 1 1 n n n n i i i i i i i i x x n x x x nx s n n n = = = = - - - = = = - - -
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Sample Statistics
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Boxplots Used to represent quartiles of data and provides a picture of the distribution
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Coefficient of Variation Measures relative variability CV = s/x for sample data Allows comparing different random variables on equal footing For a distribution with known parameters, CV = σ/μ Binomial CV = Poisson CV = Exponential CV = 1 Gamma CV = ( 29 1 / p np - 1/ λ 1/ k
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Chapter 7 Statistical Estimation and Sampling Distributions
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Parameters and Statistics A parameter is a true, fixed value that, in statistical analysis is usually unknown May be the true mean μ or true variance σ 2 of a probability distribution The true value of a parameter is estimated from sample data A statistic is a property of sample data taken from a population A point estimate of some unknown parameter is a statistic that provides a best guess at the parameter value
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Parameters and Statistics
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Point Estimates A point estimate is unbiased if The bias of a biased estimate is Suppose we don’t know the probability of success p in a sequence of Bernoulli trials If there are n trials, we use the point estimate for p The number of successes, X ~ B(n, p), so that E(X) = np, implying Therefore is an unbiased estimator of p ˆ θ ( 29 ˆ E = ( 29 ˆ E - ˆ / p X n = ( 29 ( 29 ˆ / E p E X n p = = ˆ p
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Point estimates If X 1 , …, X n are sample observations from a distribution with mean μ, consider the sample mean Observe that This shows that = X is an unbiased estimator of μ 1 / n i i X X n = = ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1/ 1/ / n n i i i i E X n E X n E X n n μ = = = = = = ˆ
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APME_2 - Applied Probability Methods for Engineers Slide...

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