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cs3_1 (1)

# cs3_1 (1) - Probability and Statistics Cumulative...

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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 ² = cov ( X;Y ) ° X ° Y Means, Variances and Covariances of Sums of Random Variables E ( a + bX + cY ) = a + X + 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 ² iid ° ° X ; ± 2 X ± 1

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³ 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 closer±to a as n ! 1 : Formally, S n converges in probability to a if for every " > 0 Pr ( j S n ± a j > " ) ! 0 as n ! 1 ³ We say that ^ ³ n (or, for brevity, ^ ³ ) is a consistent estimator for a parameter ³ if the estimator ^ ³ n converges in probability to ³ when n ! 1 ; that is, ^ ³ n p ! ³ or plim n !1 ^ ³ n = ³: ³ Law of Large Numbers. Let X 1 ; X 2 ; ::; X n be iid random variables with mean ° X and var ( X i ) = ± 2 X < 1 : Then X is a consistent estimator of ° X ; that is ° X p ! ° X : ³ Convergence in Distribution . We say that S n converges in distribution to S , and we write S n d ! S; if distribution of S n becomes °close±to the distribution of S as n ! 1 ³ Central Limit Theorem. If X 1 ; :::; X n are iid ° ° X ; ± 2 X ± with 0 < ± 2 X < 1 , then X ± ° X ° X p n = p n X ± ° X ± X d ! N (0 ; 1) Note that this also implies that p n ° ° X ± ° X ± d ! N ° 0 ; ± 2 X ± ³ Slutsky±s Theorem. Suppose that a n p ! a and S
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cs3_1 (1) - Probability and Statistics Cumulative...

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