course syllubus

# course syllubus - Probability and Statistics Cumulative...

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Unformatted text preview: Probability and Statistics Cumulative Distribution Function (CDF) of a Random Variable (RV) X CDF X ( z ) = Pr ( X & z ) Discrete: P n i =1 1 ( x i & z ) Pr ( X = x i ) ; where 1 ( E ) = 1 if E is true, and 0 otherwise. Continuous: R z &1 f ( x ) dx = R z &1 1 ( x & z ) f ( x ) dx Expectation of X Discrete Random Variable: & X = E ( X ) = x 1 Pr( X = x 1 ) + ::: + x n Pr( X = x n ) = P n i =1 x i Pr( X = x i ) Continuous Random Variable: & X = E ( X ) = R 1 &1 xf ( x ) dx Expectation of a function of X E [ g ( X )] = P n i =1 g ( x i ) Pr( X = x i ) or E [ g ( X )] = R 1 &1 g ( x ) f ( x ) dx Variance V ar ( X ) = E h ( X ¡ & X ) 2 i = ¡ 2 X V ar ( X ) = P n i =1 [ x i ¡ & X ] 2 Pr( X = x i ) or V ar ( X ) = R 1 &1 ( x ¡ & X ) 2 f ( x ) dx V ar ( X ) = E ( X 2 ) ¡ ( & X ) 2 r th moment E ( X r ) = P n i =1 x r i Pr( X = x n ) ; or E ( X r ) = R 1 &1 x r f ( x ) dx Marginal Probability P ( Y = y ) = n P i =1 P ( X = x i ;Y = y ) and Marginal Density R 1 &1 f ( y;x ) dx Conditional Probability P ( Y = y j X = x ) = P ( X = x;Y = y ) P ( X = x ) and Conditional Density f ( y;x ) f ( x ) Conditional Expectation E [ Y j X = x ] = P n i =1 y i P ( Y = y i j X = x ) or R 1 &1 yf ( y j X = x ) dy Law of Iterated Expectations E [ E ( Y j X )] = E [ Y ] De&nitions of Independence P ( Y = y j X = x ) = P ( Y = y ) or P ( X = x;Y = y ) = P ( X = x ) P ( Y = y ) Covariance and Correlation cov ( X;Y ) = E [( X ¡ & X )( Y ¡ & Y )] = ¡ XY cov ( X;Y ) = P n i =1 P m j =1 ( x i ¡ & X )( y j ¡ & Y ) P ( X = x i ;Y = y j ) cov ( X;Y ) = R 1 &1 R 1 &1 ( x ¡ & X )( y ¡ & Y ) f ( x;y ) dxdy ¢ = cov ( X;Y ) & X & Y Means, Variances and Covariances of Sums of Random Variables E ( a + bX + cY ) = a + b& X + c& Y var ( a + bX + cY ) = b 2 ¡ 2 X + 2 bc¡ XY + c 2 ¡ 2 Y cov ( a + bX + cY;d + eZ + fW ) = be¡ XZ + bf¡ XW + ce¡ Y Z + cf¡ Y W E ( XY ) = ¡ XY + & X & Y Some useful estimators Sample Mean: & X = 1 n P n i =1 X i : Sample Variance: s 2 X = 1 n & 1 P n i =1 & X i ¡ & X ¡ 2 : Sample Covariance: s XY = 1 n & 1 P n i =1 ( X i ¡ & X )( Y i ¡ Y ) note that P n i =1 ( X i ¡ & X )( Y i ¡ Y ) = P n i =1 ( X i...
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## This note was uploaded on 08/02/2011 for the course ECON 139 taught by Professor Alessandrotarozzi during the Spring '08 term at Duke.

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course syllubus - Probability and Statistics Cumulative...

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