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Unformatted text preview: Chapter 7 summary Properties of Expectation Expectation of a function of a pair of random variables: Recall from Chapters 4 and 5 that if X is a random variable, then its expectation is given by E [ X ] = x xp X ( x ) if X is discrete R - x f X ( x ) dx if X is continuous and if g ( x ) is any real-valued function E [ g ( X )] = x g ( x ) p X ( x ) if X is discrete R - g ( x ) f X ( x ) dx if X is continuous We have analogous formulas for functions of two random variables. If X, Y are any two random variables and g ( x, y ) a real-valued function of two variables, then E [ g ( X, Y )] = x,y g ( x, y ) p X,Y ( x, y ) if X and Y are discrete R - R - g ( x, y ) f X,Y ( x, y ) dydx if X and Y are continuous Expectation, variance and independence: If X and Y are independent random variables, then E [ XY ] = E [ X ] E [ Y ] and V ar ( X + Y ) = V ar ( X ) + V ar ( Y ) . Note that neither one of these formulas is true in general without the assumption of independence. Expectation of sums: Let X 1 , . . . , X n be random variables (independence not assumed). Let X = n i =1 X i . Then E [ X ] = n X i =1...
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- Spring '08