Calculate the variance of the joint probability. {Actuarial Problem, Exam P. Help!}

Question

I'm studing for exam P, and I found the following problem:

I don't know how to solve it, and I will really apreciate if someone know how to do it, and learn from it.

Thanks comunity again.

Answer 1

$Var[0.5(X+Y)] = 0.5^2 Var(X+Y) = 0.5^2[ Var(X)+Var(Y) + 2Cov(X,Y)]$

If that seems like a lot of work, you can try this

$Var[X+Y] = E[(X+Y)^2] - [E(X+Y)]^2$

and use the fact that $E[g(X,Y)] = \int_x \int_y~ g(x,y) f(x,y)~dy~dx$