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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 & X ) Y i = P n i =1 X i ( Y i Y ) Sample Correlation: r XY = s XY s X s Y Independent and Identically Distributed ( iid ) random variable. X 1 ;X 2 ;:::;X n are iid random variables if E ( X i ) = & X and V ar ( X i ) = 2 X 8 (that is, for all) i then we can write X i iid & & X ; 2 X 1 & Convergence in Probability. Loosely speaking, we say that the random variable S n converges in probability to a; or S n p ! a or, equivalently, plim n !1 S n = a if S n becomes &closer and closerto a as n ! 1 : Formally, S n converges in probability to a if for every " > Pr ( j S n a j > " ) ! as n ! 1 & We say that ^ & n (or, for brevity, ^ & ) is a consistent estimator for a parameter & if the estimator ^ & n converges in probability to & when n ! 1 ; that is, ^ & n p ! & or plim n !1 ^ & n = &: & Law of Large Numbers. Let X 1 ;X 2 ;::;X...
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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.
 Spring '08
 ALESSANDROTAROZZI
 Econometrics

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