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PropExpectation-6

# PropExpectation-6 - Welcome Back Our Friend Expectation Let...

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1 Welcome Back Our Friend: Expectation Recall expectation for discrete random variable: Analogously for a continuous random variable: Note: If X always between a and b then so is E[X] More formally: x x p x X E ) ( ] [ dx x f x X E ) ( ] [ b X E a b X a P ] [ 1 ) ( then if Generalizing Expectation Let g ( X , Y ) be real-valued function of two variables Let X and Y be discrete jointly distributed RVs: Analogously for continuous random variables:  y x Y X y x p y x g Y X g E ) , ( ) , ( )] , ( [ , dy dx y x f y x g Y X g E y x Y X ) , ( ) , ( )] , ( [ ,   Expected Values of Sums Let g(X, Y) = X + Y. Compute E[g(X, Y)] = E[X + Y] Generalized: Holds regardless of dependency between X i ’s dy dx y x f y x Y X E y x Y X ) , ( ) ( ] [ ,   dy dx y x f y dx dy y x f x y x Y X x y Y X ) , ( ) , ( , ,     dy y f y dx x f x y Y x X ) ( ) (   ] [ ] [ Y E X E n i i n i i X E X E 1 1 ] [ Tie Me Up! : Bounding Expectation If random variable X ≥ a then E[X] ≥ a Often useful in cases where a = 0 But, E[X] ≥ a does not imply X ≥ a for all X = x o E.g., X is equally likely to take on values -1 or 3. E[X] = 1. If random variables X ≥ Y then E[X] ≥ E[Y] X ≥ Y X – Y ≥ 0 E[X – Y] ≥ 0 Note: E[X Y] = E[X] + E[-Y] = E[X] E[Y] Substituting: E[X] – E[Y] ≥ 0 E[X] ≥ E[Y] But, E[X] ≥ E[Y] does not imply X ≥ Y for all X = x, Y = y ] [ 1 ) ( X E a X a P then if Sample Mean Consider n random variables X 1 , X 2 , ... X n X i are all independently and identically distributed (I.I.D.) Have same distribution function F and E[X i ] = m We call sequence of X i a sample from distribution F Sample mean: Compute is “unbiased” estimate of m n i i n X X

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