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Unformatted text preview: EECS 501 EXPECTATION Fall 2001 DEF: The expectation = expected value = mean = 1 st moment of rv x is E [ x ] = ¯ x = R Xf x ( X ) dX (continuous rv); ∑ Xp x ( X ) (discrete rv). Note: f x ( X )=mass density → E [ x ]=center of mass. 2 3 rule for linear f x ( X ). 1. E [ · ] is a linear operator: E [ ax + by ] = aE [ x ] + bE [ y ] since E [ ax + by ] = R R ( aX + bY ) f x,y ( X, Y ) dX dY = a R R Xf x,y ( X, Y ) dX dY + b R R Y f x,y ( X, Y ) dX dY = a R Xf x ( X ) dX + b R Y f y ( Y ) dY . QED. 2. Can have p x ( E [ x ]) = 0. Consider: Flip a fair coin. x=#heads. E[x]=1/2, but p x ( E [ x ]) = Pr [ x = E [ x ]] = Pr [1 / 2 head ] = 0! 3. E [ g ( x )] = R Y f y ( Y ) dY = R g ( X ) f x ( X ) dX (note f y ( Y ) dY = f x ( X ) dX ). Note: E [ g ( x )] 6 = g ( E [ x ]), e.g., E [ 1 x ] 6 = 1 /E [ x ]! Very common mistake! 4. Pr [ x > E [ x ]] 6 = Pr [ x < E [ x ]] unless f x ( X ) symmetric about X = E [ x ]. DEF: Median m of rv x is m s.t. F x ( m ) = Pr [ x ≤ m ] = Pr [ x > m ] = 1 2 . Conditional Expectations: 1. E [ x  A ] = R Xf x  A ( X  A ) dX = R R A X f x,y ( X,Y ) Pr [ A ] dX dY is a number . 2. E [ x  y = Y ] = R Xf x  y ( X  Y ) dX is a function of Y . 3. E [ x ] = R E [ x  y = Y ] f y ( Y ) dY since = R R Xf x  y ( X  Y ) dX f y ( Y ) dY = R R Xf x,y ( X, Y ) dX dY = R Xf x ( X ) dX . Iterated expectation....
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.
 Spring '10
 Lehman

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