Two research questions

I am wondering what the answers to the following questions are. I think I can work out the first one (I know it is not ANOVA), at least its first part (a). The second is more interesting, and, perhaps, could be more widely applicable.

1) Expected variance
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Given a set S and its subsets T_i (i = 1 ... k), and the corresponding variances var(S) and var(T_i), for a random subset T of S, what is p(var(T) >= Theta) assuming
(a) x \in S is distributed according to a Gaussian distribution (\mu, \Sigma)?
(b) non-parametric distribution following the observations.


2) Max-min analogue of integral images:
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Find the minimum integer value of k such that k-max chains at all pixel locations are sufficient for an O(k) time determination of a "random-access" subset max (or min)?

I could not find anything in the literature that is relevant to these. I would appreciate if