This question belongs to a problem set of MIT OCW probabilistic systems analysis and applied probability.
Random variables X and Y have the joint PMF(probability mass function)
P$_{X,Y}$ (x, y)=$\frac1 {72}$(x$^2$ + y$^2$), if x ∈ {1, 2, 4} and y ∈ {1, 3},
P$_{X,Y}$ (x, y) = 0, otherwise.
Ques. What is var(X+Y)? (var denotes variance)
My doubt: When I do it using this formula E[(X+Y)$^2$]−(E[X+Y])$^2$ , I get answer as $\frac {2188} {72} -\frac {256} 9= \frac {35} {18}$. This matches with solution provided.
But when I solve it using this formula E[(X+Y - $\mu_{X+Y}$)$^2$], I get answer as $\frac {115} {72}$. -->where $\mu_{X+Y}$ denotes expectation of (X+Y)
I have checked multiple times for calculation mistakes. I believe I have made logical error that is why I am asking here.
This is my solution using this formula E[(X+Y - $\mu_{X+Y}$)$^2$] :-
(at the end answer comes out to be $\frac {1035} {9*72}$ )
What errors have I made?
