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Math Help - Do these texts contradict eachother?

  1. #1
    Member billym's Avatar
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    Do these texts contradict eachother?


    Do these two texts contradict each other?

    Or does it mean that not all cases where E[Y | X] = E[Y] are independent, but if X and Y are independent, then E[Y | X] = E[Y] ?
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  2. #2
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    Quote Originally Posted by billym View Post
    Do these two texts contradict each other?
    No, they do not

    Or does it mean that not all cases where E[Y | X] = E[Y] are independent,
    Unless we know that X and Y are independent, E[Y | X] = E[Y] doesn't prove they are independent. E[Y | X] = E[Y] could be just a coincidence.


    but if X and Y are independent, then E[Y | X] = E[Y] ?
    Yes, if X and Y are independent, X doesn't effect Y, and vice versa. Thefore E[Y | X] = E[Y]
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  3. #3
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    Quote Originally Posted by billym View Post

    Or does it mean that not all cases where E[Y | X] = E[Y] are independent, but if X and Y are independent, then E[Y | X] = E[Y] ?
    Yes, it means what you've just written.
    It there would be this equivalence :

    for any [some conditions over] f, E[f(Y)|X]=E[Y] <=> X and Y are independent.

    So they do not contradict each other
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