Unizor - Probability - Random Variables - Expectation Sum

Our goal in this lecture is to prove that expectation of a sum of two random variables equals to a sum their expectations.
Consider the following two random experiments (sample spaces) and random variables defined on their elementary events.

Ω1=(E1,E2,...,Em )
with corresponding measure of probabilities of these elementary events
P=(P1,P2,...,Pm )
(that is, P(Ei )=Pi - non-negative numbers with their sum equaled to 1)
and random variable ξ defined for each elementary event as
ξ(Ei) = Xi where i=1,2,...m

Ω2=(F1,F2,...,Fn )
with corresponding measure of probabilities of these elementary events
Q=(Q1,Q2,...,Qn )
(that is, Q(Fj )=Qj - non-negative numbers with their sum equaled to 1)
and random variable η defined for each elementary event as
η(Fj) = Yj where j=1,2,...,n

Separately, the expectations of these random variables are:
E(ξ) = X1·P1+X2·P2+...+Xm·Pm
E(η) = Y1·Q1+Y2·Q2+...+Ym·Qn

Let's examine the probabilistic meaning of a sum of two random variables defined on two different sample spaces.
Any particular value Xi+Yj is taken by a new random variable ζ=ξ+η defined on a new combined sample space Ω=Ω1×Ω2 that consists of all pairs of elementary events (Ei,Fj ) with the corresponding combined measure of probabilities of these pairs equal to
R(Ei,Fj ) = Rij
where index i runs from 1 to m and index j runs from 1 to n.

Thus, we have defined a new random variable ζ=ξ+η defined on a new sample space Ω of M·N pairs of elementary events from two old spaces Ω1 and Ω2 as follows
ζ(Ei,Fj ) = Xi+Yj
with probability Rij
Consider a sum Ri1+Ri2+...+Rin. It represents a probability of the first experiment resulting in a fixed elementary event ei while the second experiment resulting in either F1 or in F2, or in any other elementary event it may. That is, the result of the second experiment is irrelevant and this sum simply represents a probability of the first experiment resulting in Ei, that is it is equal to Pi:
Ri1+Ri2+...+Rin = Pi.
Similarly, fixing the result of the second experiment to Fj and letting the first experiment to end up in any way it may, we conclude
R1j+R2j+...+Rmj = Qj.
Keeping in mind the above properties of probabilities Rij, we can calculate the expectation of our new random variable ζ.
E(ζ) = E(ξ+η) =
= (X1+Y1)·R11+...+(X1+Yn)·R1n +
+ (X2+Y1)·R21+...+(X2+Yn)·r2n +
...
+ (Xm+Y1)·Rm1+...+(Xm+Yn)·Rmn =
(opening parenthesis, changing the order of summation and regrouping)
= X1·(R11+...+R1n ) +
+ X2·(R21+...+R2n ) +
...
+ Xm·(Rm1+...+Rmn ) +
+ Y1·(R11+...+Rm1 ) +
+ Y2·(R12+...+Rm2 ) +
...
+ Yn·(R1n+...+Rmn ) =
(using the properties of sums of combined probabilities Rij )
= X1·P1 +...+ Xm·Pm +
+ Y1·Q1 +...+ Yn·Qn =
= E(ξ) + E(η)
End of proof.