Does correlation affect the expectation value of a sum?

[aaaa202]

We have for two random variables X and Y (one sum runs over j and one over k):

E(X+Y) = ƩƩ(x<sub>j+yk)P(X=xk,Y=yk)
= ƩƩxjP(X=xk,Y=yk) + ƩƩykP(X=xk,Y=yk)

Now this can be simplified to obtain E(X+Y)=E(X)+E(Y) if we use that:
P(X=xk,Y=yk) = P(X=xk)P(Y=yk), because then (and same goes the other way around):
ƩjP(Y=yk)P(X=xj)= P(Y=yk)

But all this requires X and Y to be uncorrelated. Does the derivation above also hold if generally:

P(xj,yk)=P(xj l Y=yk)P(Y=yk)?</sub>

 

[Stephen Tashi]

Which indexes are varying on each of your summation signs isn't clear.

I suggest you rewrite your question using the forums LaTex. (See https://www.physicsforums.com/showpost.php?p=3977517&postcount=3)

Calculating the expected value of the sum of two variables using their joint distribution does not depend on whether the two variables are independent or not. "Correlation" is also not relevant.

 

[statdad]

Work with your original double sum - you should find the result of interest is true in each case (as long as expectations exist for your distributions)