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Conditional Expectation of Sum

  1. Aug 10, 2011 #1
    Hi everyone,

    I have a feeling the following property is true but I can't find it stated in any textbook/online reference. Maybe it's not true... Can someone verify/disprove this equation?

    [itex]E(A+B|C) = E(A|C) + E(B|C)[/itex]
  2. jcsd
  3. Aug 10, 2011 #2


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    Science Advisor

    Is this a homework question?

    Regardless of your answer, what do you know about the definition of expectation and in particular conditional expectation?
  4. Aug 10, 2011 #3
    If you mean E((A+B)|C) by E(A+ B|C) , yes.
  5. Aug 10, 2011 #4
    It's not a homework question.

    By definition of conditional expectation, we have in the discrete case
    [itex]E(A|C=c) = \sum_{a} a P(A=a|C=c)[/itex]

    [itex]E(B|C=c) = \sum_{b} b P(B=b|C=c)[/itex]

    [itex]E((A+B)|C=c) = \sum_{a,b} (a+b) P(A=a,B=b|C=c)[/itex]

    It doesn't seem like the sum of the first two should equal the last. But maybe my sum formula for the last one is wrong.
  6. Aug 11, 2011 #5

    Stephen Tashi

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    We should be able to make progress in simplifying:
    because proving [itex] E(A+B) = E(A) + E(B) [/itex] would involve dealing with a similar equation.

    [itex] \sum_{a,b}(a+b) P(A=a,B=b|C=c) = \sum_{a,b}a P(A=a,B=b|C=c) + \sum_{a,b} b P(A=a,B=b|C=c) [/itex]

    [itex] = \sum_a \sum_b a P(A=a,B=b|C=c) + \sum_a \sum_b b P(A=a,B=b|C=c) [/itex]

    [itex] = \sum_a a \sum_b P(A=a,B=b|C=c) = \sum_b b \sum_a P(A=a,B=b|C=c) [/itex]

    [itex] = \sum_a a P(A=a|C=c) + \sum_b b P(B=b|C=c) [/itex]
  7. Aug 11, 2011 #6
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