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Unformatted text preview: Probability Plots; QQ . . . Heavy tails? Title Page JJ II J I Page 1 of 23 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 11: Probability Plotting; Joint Densities David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA [email protected] February 25, 2009 Probability Plots; QQ . . . Heavy tails? Title Page JJ II J I Page 2 of 23 Go Back Full Screen Close Quit 1. Probability Plots; QQplots. Problem: • Bunch of numbers z 1 , . . . , z n dropped on your desk. • Hope these represent a sample from some population with a numerical characteristic such that the percent age of the population with the characteristic can be described by f ( x ). • In other words : hope the numbers are realizations of iid rv’s Z 1 , . . . , Z n with common density f ( x ) and dis tribution function F ( x ). • How do we guess what f ( x ) is?? – Histograms – Probability plot or QQplot Probability Plots; QQ . . . Heavy tails? Title Page JJ II J I Page 3 of 23 Go Back Full Screen Close Quit Prelude: Sample cdf Suppose Z 1 , . . . , Z n iid rv’s ∼ F ( x ). Fix x and define the random quantity called the empirical (cumulative) distri bution function or sample distribution function ˆ F n ( x ) = 1 n n X i =1 1 [ Z i ≤ x ] = % of the X ’s ≤ x . For fixed x , n ˆ F n ( x ) ∼ b ( k ; n, p = F ( x ) ) . So as with the binomial discussion of the CLT ˆ F n ( x ) d ≈ F ( x ) + c X , 1 √ n . So • ˆ F n ( x ) is close to F ( x ) for each x . • Hope ˆ F ← n ( p ) is close to F ← ( p ). Probability Plots; QQ . . . Heavy tails? Title Page JJ II J I Page 4 of 23 Go Back Full Screen Close Quit BUT: • F ← ( p ) are the percentiles of the distribution F . • ˆ F n ( x ) is not continuous, strictly increasing so how do we define ˆ F ← n ( p )? Probability Plots; QQ . . . Heavy tails? Title Page JJ II J I Page 5 of 23 Go Back Full Screen Close Quit Figure 1: CDF for 3 points. Probability Plots; QQ . . . Heavy tails? Title Page JJ II J I Page 6 of 23 Go Back Full Screen Close Quit The probability plot or QQ plot: Roughly, we plot ( ˆ F ← n ( p ) , F ← ( p ) ) , < p < 1 . If we have guessed correctly about the true F , this should be approximately a straight line....
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This note was uploaded on 03/05/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell.
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