# The mean of a sum of variables.

How do I prove that the mean of a random variable Z which is the sum of to other random variables X and Y is the sum of the mean of X with the mean of Y?

## Answers and Replies

mathman
Science Advisor
Essentially the theorem is equivalent to the theorem that the integral of a sum is the sum of the integrals.

well I obviously know that the integral of the sum is the sum of the integral but I don't know how I can relate that to the situation a mentioned, can you please be more specific?
I'm trying to prove it and I'm getting a convultion integral so far...
thank you.

well I obviously know that the integral of the sum is the sum of the integral but I don't know how I can relate that to the situation a mentioned, can you please be more specific?
I'm trying to prove it and I'm getting a convultion integral so far...
thank you.

Yes that'll eventually give you a proof for the special case where the rv's are independent and have densities (involves reversing the order of integration and a change of variables).

Another approach that would work for the non-independent case is to consider separately the joint distribution and marginal distributions of X and Y.

well but I'm not being able to prove it either for dependent or independent variables, can you please show me the proof or tell me where I can find it? thank you

mathman
Science Advisor
Two random variables.
E(X+Y)=∫∫(x+y)dF(x,y)=∫∫xdF(x,y) + ∫∫ydF(x,y).
Integrate with respect to y in the first integral and integrate with respect to x in the second integral. You will be left with E(X) + E(Y).

In the above F(x,y) is the joint distribution function.