I would like to find a computationally quick and sufficiently accurate way (i.e., either analytically or numerical approximation) to compute a conditional probability for a specific multi-variate normal distribution, which is defined as follows:

Let x1, ..., xn be n normal I.I.D. (univariate) random variables. And let A1, ..., Ak be defined as sums of subsets of those random variables, i.e.,:

For each i: Ai = \sum_{j \in S} a_j, where S \subseteq {x1, ..., xn}.

I am now looking for the conditional probabilities of Ai:

(1) Prob ( Ai <= t | Aj <= t, where j=1,...,n and j<>i )

and

(2) Prob ( Ai <= X | Aj <= t, where j=1,...,n and j<>i ), where X is any number smaller than t.

To the best to my knowledge, the conditional density of a multi-variate normal density is also a (uni-variate) normal distribution. Is that right? In this case, of course, ideally I would like to find the entire density of this conditional probability in form N(mean, variance).

I tried via Monte-Carlo simulation, but approximation is not sufficiently accurate when using 500,000 samples, 2000 bins (to classify the results) for a "problem" with 50 random variables, i.e. n=50.

Analytically that doesn't seem to be trivial. I have asked quite a few Statisticians around me and so far we haven't found a solution.

Any hint is appreciated.

Thanks,

SDJ