  It is known that random monic integral polynomials of bounded degree $d$ and
integral coefficients distributed uniformly and independently in $[-H,H]$ are
irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$.
In this paper, we give a general criterion for guaranteeing the same conclusion
under much more general coefficient distributions, allowing them to be
nonuniformly and dependently distributed over arbitrary sets.
