I have seen this line in a computation and I wonder if that makes sense : E ( X | Y ) = ∫ E ( X | Y = y ) d P ( Y = y ) where P is a probability measure, X , Y are random variables, so is E ( X | Y ). But I'm not familiar with this notation. If this is right, how can I understand it ?

Question

I have seen this line in a computation and I wonder if that makes sense :       E    (  X      |    Y  )  =  ∫      E    (  X      |    Y  =  y  )  d  P  (  Y  =  y  ) where P is a probability measure,   X  ,  Y are random variables, so is       E    (  X      |    Y  ). But I&#039;m not familiar with this notation. If this is right, how can I understand it ?

Kayley Jackson · Accepted Answer

You have that       E    (  X      |    Y  ) is, by definition,   σ  (  Y  )-measurable, and it can be shown that there is a measurable function   h  :      R    →      R   such that       E    (  X      |    Y  )  =  h  (  Y  ).Clearly   h  (  Y  )  d  P  ≪  d  P, and so   h  (  y  )  d      P    Y    (  y  )  ≪  d      P    Y    (  y  ), where   d      P    Y   is the measure induced by   Y in       R  . Then we set       E    (  X      |    Y  =  y  )  :=  h  (  y  ). However, your identity is false, according to the above we have that      ∫                  R                  E    (  X      |    Y  =  y  )  d      P    Y    (  y  )  =      ∫          Ω            E    (  X      |    Y  )  d  P  =      ∫          Ω        X  d  P  =      E    (  X  )

I have seen this line in a computation and I wonder if that makes sense : <mrow class="MJX-TeXAt

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