Conditional Expectation of Sum

  • Thread starter CantorSet
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  • #1
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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]
 

Answers and Replies

  • #2
chiro
Science Advisor
4,790
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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]
Is this a homework question?

Regardless of your answer, what do you know about the definition of expectation and in particular conditional expectation?
 
  • #3
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If you mean E((A+B)|C) by E(A+ B|C) , yes.
 
  • #4
44
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Is this a homework question?

Regardless of your answer, what do you know about the definition of expectation and in particular conditional expectation?
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.
 
  • #5
Stephen Tashi
Science Advisor
7,392
1,367
We should be able to make progress in simplifying:
[itex]E((A+B)|C=c) = \sum_{a,b} (a+b) P(A=a,B=b|C=c)[/itex]
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]
 
  • #6
44
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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]
Thanks.
 

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