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Unformatted text preview: [ X + c ] = VAR [ X ] VAR [ cX ] = c 2 VAR [ X ] • The n th moment of the rv X is deﬁned by E [ X n ] = Z ∞∞ x n f X ( x ) dx The mean and variance can be seen to be deﬁned in terms of the ﬁrst two moments, E [ X ] and E [ X 2 ] . Of greater signiﬁcance are the central moments E [( XE [ X ]) n ] = Z ∞∞ ( xE [ X ]) n f X ( x ) dx Example Find the variance of a Gaussian random variable. Conditional Expected Values Expected values with respect to conditional probability density functions are called conditional expected values . If M is a conditioning event, E [ g ( X )  A ] = Z ∞∞ g ( x ) f X ( x  A ) dx is the conditional expected value of g ( X ) , given A . The conditional variance of a random variable X with respect to a conditioning event A is VAR [ X  A ] = E [ X 2  A ]E [ X  A ] 2...
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 Fall '08
 DASILVER
 Variance, Probability theory, probability density function

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