Lect13_2700_s09 - ENGRD 2700 Basic Engineering Probability...

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Statistics Sums and averages Expectation Normal Title Page JJ II J I Page 1 of 26 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 13: Multivariate Distributions; Statistics and their Distributions David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA [email protected] March 4, 2009
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Statistics Sums and averages Expectation Normal Title Page JJ II J I Page 2 of 26 Go Back Full Screen Close Quit 1. Statistics and Distributions Hypothetical setup: There is a probability model ( S, { Events } , P ) and defined on this model is a random vector X = ( X 1 , . . . , X n ) . (Reminder: Each X i is a function on S with range a subset of R .) Probabilities about the random vector are specified by either a probability mass function p ( x 1 , . . . , x n ) (discrete case) or a probability density function f ( x 1 , . . . , x n ) (continuous case). In nice (but certainly not all) cases X 1 , . . . , X n are iid which statisticians call a random sample .
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Statistics Sums and averages Expectation Normal Title Page JJ II J I Page 3 of 26 Go Back Full Screen Close Quit A probability model is a model of an experiment before it is carried out. (Before we flip a coin, we have an idea of the possible outcomes and the probabilities of these outcomes. But we do not know with certainty what will happen.) Now carry out the experiment and observe values of X 1 , . . . , X n . This observational study gives us data , that is n real numbers x 1 , . . . , x n . (If we were flipping a coin n times, the result of the experiement would be a sequence of length n consisting of 0’s and 1’s.)
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Statistics Sums and averages Expectation Normal Title Page JJ II J I Page 4 of 26 Go Back Full Screen Close Quit This point of view is we have a model and we observe this model to get data. The statistician’s point of view is the reverse: We have data. We hope there is a model which produced the data. From the data, we have to guess the model and assess the risk of our guess. To help us decide on the correct model, we use func- tions of the data called statistics .
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Statistics Sums and averages Expectation Normal Title Page JJ II J I Page 5 of 26 Go Back Full Screen Close Quit Statistics Definition: A statistic is a (known) function g ( x 1 , . . . , x n ) of the data. Note g : { ( y 1 , . . . , y n ) : each y i is a possible value of X i } 7→ R . Examples: Sample mean: g ( x 1 . . . , x n ) = ¯ x = x 1 + · · · + x n n . Sample variance: g ( x 1 . . . , x n ) = s 2 = 1 n - 1 n X i =1 ( x i - ¯ x ) 2 . Sample median: middle value of the x ’s after ordering. Sample range: largest x - smallest x .
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