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background_overhead - Prob Stat 1 PROBABILITY AND...

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Prob & Stat 1 PROBABILITY AND STATISTICS Some Basic Definitions Random variable large set of possible values only one will occur The set of possible values and their probabilities = the probability distribution
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Prob & Stat 2 Continuous random variable has a probability density function (pdf) f X such that P ( X A ) = Z A f X ( x ) dx for all sets A
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Prob & Stat 3 CDFs cumulative distribution function (CDF) of X is F X ( x ) := P ( X x ) If X has a pdf then F X ( x ) := Z x -∞ f X ( u ) du
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Prob & Stat 4 Quantiles if the CDF of X is continuous and strictly increasing then it has a inverse function F - 1 for q between 0 and 1, F - 1 ( q ) is called the q th quantile or 100 q th percentile
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Prob & Stat 5 probability X is below its q th quantile is q : P { X F - 1 ( q ) } = q also called the lower quantile the q th upper quantile is the 1 - q th lower quantile 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 q F -1 (q) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 q F -1 (q) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 q F -1 (q)
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Prob & Stat 6 median is the 50% percentile or .5 quantile 25% and 75% percentiles (.25 and .75 quantiles) are called the first and third quartiles for 95% confidence intervals we use the 0.025 and 0.975 quantiles, i.e., the 0.025 lower and 0.025 upper quantiles.
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Prob & Stat 7 Expectations and Variances — The expectation of X is E ( X ) := Z + -∞ xf X ( x ) dx variance of X is σ 2 X := Z { x - E ( X ) } 2 f X ( x ) dx = E { X - E ( X ) } 2 Useful formula: σ 2 X = E ( X 2 ) - { E ( X ) } 2 standard deviation is the square root of the variance: σ X := p E { X - E ( X ) } 2
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Prob & Stat 8 If X 1 , . . . , X n is a sample from a prob dist’n, then expectation estimated by sample mean X = n - 1 n X i =1 X i the variance estimated by sample variance s 2 X = n i =1 ( X i - X ) 2 n - 1
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Prob & Stat 9 Correlation and Covariance σ XY = E { X - E ( X ) }{ Y - E ( Y ) } If ( X, Y ) are continuously distributed, then σ XY = Z { x - E ( X ) }{ y - E ( Y ) } f XY ( x, y ) dx dy
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Prob & Stat 10 Useful formulas: σ XY = E ( XY ) - E ( x ) E ( y ) σ XY = E [ { X - E ( X ) } Y ] σ XY = E [ { Y - E ( Y ) } X ] σ XY = E ( XY ) if E ( X ) = 0 or E ( Y ) = 0
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Prob & Stat 11 Correlation coefficient between X and Y : ρ XY := σ XY X σ Y for any ( X, Y ) it is true that - 1 ρ XY 1
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Prob & Stat 12 Given a bivariate sample { ( X i , Y i ) } n i =1 , the sample correlation coefficient is n - 1 n i =1 ( X i - X )( Y i - Y ) s X s Y (1) where X and Y are the sample means s X and s Y are the sample standard deviations
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Prob & Stat 13 -5 0 5 -3 -2 -1 0 1 2 3 r = 0.5 -5 0 5 -3 -2 -1 0 1 2 3 r = 0.25 -5 0 5 -3 -2 -1 0 1 2 3 r = 0.95 -5 0 5 -2 -1 0 1 2 3 4 r = 0.11 -5 0 5 -4 -2 0 2 4 6 r = 0.83 -5 0 5 -3 -2 -1 0 1 2 3 r = 0.01 -5 0 5 -4 -3 -2 -1 0 1 2 3 r = -0.89 -5 0 5 -3 -2 -1 0 1 2 3 r = -1
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Prob & Stat 14 an absolute correlation of .25 is very weak an absolute correlation of .5 is only moderate an absolute correlation of .95 is rather strong an absolute correlation of 1 implies a linear relationship a strong nonlinear relationship may or may not imply a high correlation
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Prob & Stat 15 positive correlations increasing relationship negative correlations decreasing relationship
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Prob & Stat 16 X and Y are independent if for all sets A and B , P ( X A and Y B ) = P ( X A ) P ( Y B ) .
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