johnqwertyful said:
I am only given numbers, no other information.
Well, then, all you can do is either make an assumption (such as independence of lists and equal probabilities on all list members) or else do what I suggested above. For example, to minimize EX, just put probability = 1 on the smallest x-value and zero probability elsewhere; to maximize EX, put all the probability on the largest x-value. To minimize Var(X) just put all the probability on any single x-value; that will make Var(X) = 0.
The maximization of Var(X) is more challenging, and I will leave it to you. It is a quadratic optimization problem in the probabilities ##p_1, p_2, \ldots, p_8## on the eight listed x-values (assuming X list has 8 entries). You want to choose ##\{ p_i \}## to maximize
V(p) = E(X^2) - (EX)^2 = \sum_{i=1}^8 x_i^2 p_i - \sum_{i=1}^8 \sum_{j=1}^8 x_i x_j p_i p_j,
subject to the constraints ##p_i \geq 0, i=1,\ldots, 8## and ##\sum_{i=1}^8 p_i = 1##. This is a quadratic optimization problem in non-negative variables ##p_i##, subject to a single linear equality constraint. (Here, I assume the ##x_i## are just some given, known numbers.)
Similarly, you can look at the problems of max or min ##\text{Cov}(X,Y)## or ##\text{Var}(X+Y)## as a quadratic optimization problem involving the 80 variables ##h(i,j), \: i = 1,\ldots,8 \: j = 1, \ldots,10.## You need all ##h(i,j) \geq 0## and ##\sum_i \sum_j h(i,j) = 1.## Such problems can be handled numerically using, for example, the EXCEL Solver. Alternatively, you can think harder about the problem and try to get some properties of a solution. Be warned, though, that in such cases there can be a significant difference between the minimization and maximization problems; one of these may be relatively easy and the other very hard (for example, needing global optimization methods in case there are multiple local optima, etc.)
