Lets start with n 0 i 1 yi y xi x sx sy 2 the

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Unformatted text preview: 1 n i =1 (xi = 1 n−1 n i =1 (xi = 1 n−1 n i =1 (xi ¯ ¯ − x )(y − y ) sx sy ¯ ¯ − x )([bxi + a] − [bx + a]) sx ¯ ¯ − x )(xi − x ) sign(b) sx sx 2 = sx sign(b) /sx sx = sign(b) 2 b2 sx The correlation coefficient is always between The correlation coefficient is always between -1 and 1. Let’s start with n 0≤ i =1 ¯ yi − y ¯ xi − x − sx sy 2 The correlation coefficient is always between -1 and 1. Let’s start with n 0≤ i =1 n = i =1 ¯ yi − y ¯ xi − x − sx sy ¯ xi − x sx 2 n 2 −2 i =1 ¯ ¯ xi − x yi − y + sx sy n i =1 ¯ yi − y sy 2 The correlation coefficient is always between -1 and 1. Let’s start with n 0≤ i =1 n = i =1 ¯ yi − y ¯ xi − x − sx sy ¯ xi − x sx 2 n 2 −2 i =1 ¯ ¯ xi − x yi − y + sx sy n i =1 22 22 = (n − 1)sx /sx − 2(n − 1)rxy + (n − 1)sy /sy ¯ yi − y sy 2 The correlation coefficient is always between -1 and 1. Let’s start with n 0≤ i =1 n = i =1 ¯ yi − y ¯ xi − x − sx sy ¯ xi − x sx 2 n 2 −2 i =1 ¯ ¯ xi − x yi − y + sx sy n i =1 22 22 = (n − 1)sx /sx − 2(n − 1)rxy + (n − 1)sy /sy So, 2(n − 1)rxy ≤ (n − 1)2. ¯ yi − y sy 2 What about the other direction. Let’s start with n 0≤ i =1 n = i =1 ¯ ¯ xi − x yi − y + sx sy ¯ xi − x sx 2 n 2 +2 i =1 ¯ ¯ xi − x yi − y + sx sy n i =1 22 22 = (n − 1)sx /sx + 2(n − 1)rxy + (n − 1)sy /sy So, 2(n − 1)rxy ≥ − (n − 1)2. ¯ yi − y sy 2 Chebechev’s inequality says ¯ The proportion of observations more than t from x is Chebechev’s inequality says ¯ The proportion of observations more than t from x is 2 bounded by sx /t 2 . Chebechev’s inequality says ¯ The proportion of observations more than t from x is 2 bounded by sx /t 2 . 2 sx = 1 n−1 n ¯ (xi − x )2 i =1 Chebechev’s inequality says ¯ The proportion of observations more than t from x is 2 bounded by sx /t 2 . 2 sx = 1 n−1 = 1 n−1 n ¯ (xi − x )2 i =1 ¯ (xi − x )2 + ¯ {i :|xi −x |>t } 1 n−1 ¯ (xi − x )2 ¯ {i :|xi −x |≤t } Chebechev’s inequality says ¯ The proportion of observations more than t from x is 2 b...
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This document was uploaded on 03/02/2014 for the course STATS 4150 at Barnard College.

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