Conditional Expectation of Sum

[CantorSet]

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?

$E(A+B|C) = E(A|C) + E(B|C)$

 

[chiro]

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?

$E(A+B|C) = E(A|C) + E(B|C)$

Is this a homework question?

Regardless of your answer, what do you know about the definition of expectation and in particular conditional expectation?

 

[Eynstone]

If you mean E((A+B)|C) by E(A+ B|C) , yes.

 

[CantorSet]

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
$E(A|C=c) = \sum_{a} a P(A=a|C=c)$

$E(B|C=c) = \sum_{b} b P(B=b|C=c)$

$E((A+B)|C=c) = \sum_{a,b} (a+b) P(A=a,B=b|C=c)$

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.

 

[Stephen Tashi]

We should be able to make progress in simplifying:

$E((A+B)|C=c) = \sum_{a,b} (a+b) P(A=a,B=b|C=c)$

because proving $E(A+B) = E(A) + E(B)$ would involve dealing with a similar equation.

$\sum_{a,b}(a+b) P(A=a,B=b|C=c) = \sum_{a,b}a P(A=a,B=b|C=c) + \sum_{a,b} b P(A=a,B=b|C=c)$

$= \sum_a \sum_b a P(A=a,B=b|C=c) + \sum_a \sum_b b P(A=a,B=b|C=c)$

$= \sum_a a \sum_b P(A=a,B=b|C=c) = \sum_b b \sum_a P(A=a,B=b|C=c)$

$= \sum_a a P(A=a|C=c) + \sum_b b P(B=b|C=c)$

 

[CantorSet]

We should be able to make progress in simplifying:

because proving $E(A+B) = E(A) + E(B)$ would involve dealing with a similar equation.

$\sum_{a,b}(a+b) P(A=a,B=b|C=c) = \sum_{a,b}a P(A=a,B=b|C=c) + \sum_{a,b} b P(A=a,B=b|C=c)$

$= \sum_a \sum_b a P(A=a,B=b|C=c) + \sum_a \sum_b b P(A=a,B=b|C=c)$

$= \sum_a a \sum_b P(A=a,B=b|C=c) = \sum_b b \sum_a P(A=a,B=b|C=c)$

$= \sum_a a P(A=a|C=c) + \sum_b b P(B=b|C=c)$

Thanks.