Suppose that X_1, X_2,... X_5000 are discrete and dependent non-identically distributed random variables, whose marginal distributions are known, but whose joint distribution is not known.
Is there a way to estimate bounds for the cumulative distribution function P(X_1 + X_2 + ... +X_1000 ≤ a)? Which conditions would need to be met?
I tried this with two Bernoulli variables. Let X and Y be Bernoulli with P(X=1) = $p_1$ and P(Y=1) = $p_2$ and we're looking for $P(X + Y <= n)$ where $n = 0,1,2$.
{X + Y = n} can be written as union of disjoint events {X = k,Y = n-k}, $0 \leq k \leq n$. Then $P(X + Y = n) = \sum_{k=0}^n P(X = k,Y = n-k)$ and we can compute
$P(X + Y = 0) = P(X = 0,Y = 0) = P(X = 0|Y = 0) \cdot (1-p_2)$
$P(X + Y = 1) = P(X = 0,Y = 1) + P(X = 1,Y = 0)$ $= P(X = 0|Y = 1) \cdot p_2 + P(X = 1,Y = 0)\cdot(1-p2)$
$P(X + Y = 2) = P(X=0,Y=2) + P(X=1,Y=1) + P(X=2,Y=0)$ $= 0 + P(X=1|Y=1) \cdot p_2 + 0$
I guess we could compute a number of possible joint distributions by assuming different conditional probability tables, but that doesn't seem very useful and I get stuck here.