The support of the pdf of y is 0 1 so let vo 0 1 then

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Unformatted text preview: retation, because P {X = uo } = 0 for any uo . But if the joint density function is sufficiently regular, then the conditional density is a limit of conditional probabilities of events: P Y ∈ v − 2, v + 2 X ∈ uo − 2 , uo + fY |X (v |uo ) = lim 2 . →0 Also, the definition of conditional pdf has the intuitive property that fX,Y (u, v ) = fY |X (v |u)fX (u). So (4.5) yields a version of the law of total probability for the pdf of Y : ∞ fY |X (v |u)fX (u)du. fY (v ) = (4.7) −∞ The expectation computed using the conditional pdf is called the conditional expectation (or conditional mean) of Y given X = u, written as ∞ E [Y |X = u] = vfY |X (v |u)dv. −∞ So E [Y |X = u] is a deterministic function of u. To emphasize that fact, we could denote it by g , so g (u) = E [Y |X = u] for all u. If that function is applied to the random variable X , the result is a random variable denoted by E [Y |X ] = g (X ). In summary, E [Y |X = u] is a deterministic function of u. In contrast, E [Y |X ] is obtained by applying that same deterministic function, b...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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