Concept guide
Expectation in finite models: proving linearity, with an instance of dependent variables
For a nonempty finite sample space S with nonnegative weights w_s summing to 1, prove E[c]=c, E[aX+b]=aE[X]+b and E[X+Y]=E[X]+E[Y] (X, Y on the same S, independence not assumed). Then on S={1,2,3} with w₁=1/2, w₂=1/4, w₃=1/4, X=(2,−2,0), Y=(1,3,−1), compute E[X], E[Y], E[X+Y] and E[−3X+5], show that X and Y are not independent while linearity still holds, and use E[XY]≠E[X]E[Y] to show that linearity does not give a product formula.