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# 2 sx 1 n1 1 n1 1 n1 1 n1 n xi x 2 i 1 xi

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Unformatted text preview: ounded by sx /t 2 . 2 sx = 1 n−1 = 1 n−1 ≥ 1 n−1 n ¯ (xi − x )2 i =1 ¯ (xi − x )2 + ¯ {i :|xi −x |>t } ¯ (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 bounded by sx /t 2 . 2 sx = 1 n−1 = 1 n−1 ≥ ≥ 1 n−1 1 n−1 n ¯ (xi − x )2 i =1 ¯ (xi − x )2 + ¯ {i :|xi −x |>t } ¯ (xi − x )2 ¯ {i :|xi −x |>t } t2 ¯ {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 bounded by sx /t 2 . 2 sx = 1 n−1 = 1 n−1 ≥ ≥ 1 n−1 1 n−1 n ¯ (xi − x )2 i =1 ¯ (xi − x )2 + ¯ {i :|xi −x |>t } ¯ (xi − x )2 ¯ {i :|xi −x |>t } t2 ¯ {i :|xi −x |>t } 1 ¯ ≥ t 2 #{i : |xi − x | > t } n 1 n−1 ¯ (xi − x )2 ¯ {i :|xi −x |≤t } Elements of Probability Theory With a li4le combinatorics Think about whether the “setup” comports with common sense We’ll draw some far- reaching conclusions, so you want to be sure that the foundaAons are solid. Sample Space •  All possible outcomes •  We wouldn’t have “probability” if there were only one possible outcome •  Even if we don’t know the sample space, there sAll “is” one Example of Sample Space •  Record the blood pressure of the next three people to be admi4ed to a blood pressure study. –  What do you think is the sample space? When the sample space is more than the possible outcomes. •  (Some of) the “random” factors that inﬂuence the sample space could be included as latent aspects of the outcome, and so contribute to a more complicated sample space. Events •  Subsets of the sampl...
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