**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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