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?
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.
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
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.