background_overhead - Prob & Stat 1 PROBABILITY...

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Unformatted text preview: 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 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 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 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 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 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 q F-1 (q) 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 q F-1 (q) 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 q F-1 (q) 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. Prob & Stat 7 Expectations and Variances The expectation of X is E ( X ) := Z + - xf X ( x ) dx variance of X is 2 X := Z { x- E ( X ) } 2 f X ( x ) dx = E { X- E ( X ) } 2 Useful formula: 2 X = E ( X 2 )- { E ( X ) } 2 standard deviation is the square root of the variance: X := p E { X- E ( X ) } 2 Prob & Stat 8 If X 1 ,...,X n is a sample from a prob distn, 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 Prob & Stat 9 Correlation and Covariance XY = E { X- E ( X ) }{ Y- E ( Y ) } If ( X,Y ) are continuously distributed, then XY = Z { x- E ( X ) }{ y- E ( Y ) } f XY ( x,y ) dxdy Prob & Stat 10 Useful formulas: XY = E ( XY )- E ( x ) E ( y ) XY = E [ { X- E ( X ) } Y ] XY = E [ { Y- E ( Y ) } X ] XY = E ( XY ) if E ( X ) = 0 or E ( Y ) = 0 Prob & Stat 11 Correlation coefficient between X and Y : XY := XY / X Y for any ( X,Y ) it is true that- 1 XY 1 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 Prob & Stat 13-5 5-3-2-1 1 2 3 r = 0.5-5 5-3-2-1 1 2 3 r = 0.25-5 5-3-2-1 1 2 3 r = 0.95-5 5-2-1 1 2 3 4 r = 0.11-5 5-4-2 2 4 6 r = 0.83-5 5-3-2-1 1 2 3 r = 0.01-5 5-4-3-2-1 1 2 3 r = -0.89-5 5-3-2-1 1 2 3 r = -1 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...
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background_overhead - Prob & Stat 1 PROBABILITY...

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