Bivariate Binomial Moments and Bonferroni-type Inequalities

We obtain bivariate forms of Gumbel's, Fr\'echet's and Chung's linear inequalities for $P(S\ge u, T\ge v)$ in terms of the bivariate binomial moments $\{S_{i,j}\}$, $1\le i\le k, 1\le j\le l$ of the joint distribution of $(S,T)$. At $u=v=1$, the Gumbel and Fr\'echet bounds improve monotonically with non-decreasing $(k,l)$. The method of proof uses combinatorial identities, and reveals a multiplicative structure before taking expectation over sample points.