04_02 - p. 4-11 Expectation (Mean) and Variance Q: We often...

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p. 4-11 Q : We often characterize a person by his/her height, weight, hair color, …. How can we “roughly” characterize a distribution? • Definition: If X is a discrete r.v. with pmf f X and range , then the expectation (or called expected value ) of X is provided that the sum converges absolutely. Expectation (Mean) and Variance Example. If all value in are equally likely, then E ( X ) is simply the average of the possible values of X . Example (Committees). In the committees example, Example (Indicator Function). For an event A  , the indicator function of A is the r.v. E ( X )=0 · 5 210 +1 · 50 210 +2 · 100 210 +3 · 50 210 +4 · 5 210 =2 . X 1 A ( ω )= 1 , if ω A , 0 , if ω / A . X E ( X x X xf X ( x ) , p. 4-12 Intuitive Interpretation of Expectation Expectation of a r.v. parallels the notion of a weighted average, where more likely values are weighted higher than less likely values. It is helpful to think of the expectation as the “center” of mass of the pmf center of gravity: If we have a rod with weights f X at each possible point x i then the point at which the rod is balanced is called the center of gravity. Its range is {0, 1} and its pmf is f (0)= P ( A c )=1 P ( A ) and f (1)= P ( A ), for a p.m. P defined on . So, E ( 1 A · [1 P ( A )] + 1 · P ( A P ( A ) .
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This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shao-weicheng during the Fall '11 term at National Tsing Hua University, China.

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04_02 - p. 4-11 Expectation (Mean) and Variance Q: We often...

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