E[X + Y] = E[X] + E[Y]

The expected value of the random variable X is:

Similarly, the expected value of Y is:

We can write the probabilities of X and Y as a joint distribution:

We can define the expected value like this:

Consider E(X), if you were to add all the probabilities of y_j, the value of x_i would not change throughout all those n summation. By the distributive property of summation, we get:

We can do the same for E(Y):

Now lets add E(Y) and E(X):

Lets use the associative identity of summation:

And this is equal to: