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Unformatted text preview: M4056 Lecture Notes. September 1, 2010 Section 5.2. (cont.) Reminders: i ) E ( aX + bY ) = a E ( X ) + b E ( Y ) ii ) E ( X 2 ) = Var ( X ) + E ( X ) 2 (see page 60, 2.3.1) iii ) Var ( aX + b ) = a 2 Var ( X ) (see page 60) iv ) If X and Y are independent, Var ( aX + bY ) = a 2 Var ( X ) + b 2 Var ( Y ). (Word to wise: good material for quiz questions here.) Suppose X 1 , . . ., X n iid . Suppose is the expected value of the common distribution of the X i and 2 is its variance. The two statistics of greatest importance are: The sample mean: X := 1 n n i =1 X i , The sample variance: S 2 = 1 n- 1 n i =1 ( X i- X ) 2 . The expected value and variance of the statistic X . E ( X ) = 1 n ( E ( X 1 ) + + E ( X n )) = (Theorem 5.2.6.a, p. 213-4). V ar ( X ) = 1 n 2 ( V ar ( X 1 ) + + V ar ( X n )) = 2 /n (Theorem 5.2.6.b, p. 213-4). To deduce information about S 2 (see next page), we will use a fact of arithmetic: Theorem 5.2.4. For any numbers...
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- Fall '08