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Help on expectation

  1. Mar 24, 2010 #1
    I have two random variables Y and X and Y is dependent of X, though X is not the only source of variability of Y. With fixed X=x, Y(x) follows gaussian law. X also follows gaussian law.

    In what cases can I move from
    E[ Y(X) ]

    to
    E[ Y(E(X))]

    someone has any idea?
    is there a text You recomend on the topic?
    thanks..
     
  2. jcsd
  3. Mar 27, 2010 #2
    I think you mean E(Y|X).

    Can you combine X and R where R ( for 'residual') includes all other extraneous conditions on Y? If so, then write P(X)+P(R)-P(X)P(R)=P(X') assuming X and R are independent.

    Then use Bayes' Theorem:

    P(Y|X')=P(X'|Y)P(Y)/P(X')

    The validity if this approach assumes that P(Y|X') includes all conditions on Y.

    EDIT: You could of course simply find P(Y|X) the same way, but this does not give you as good a description of the behavior of Y when you know of other conditions on Y. E(Y|X') is the value of Y|X' when P(Y|X') is maximal.
     
    Last edited: Mar 27, 2010
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