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We have for two random variables X and Y (one sum runs over j and one over k):
E(X+Y) = ƩƩ(xj+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)?
E(X+Y) = ƩƩ(xj+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)?