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.
 Spring '08
 Zahrn
 The Land

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