We obtain bivariate forms of Gumbel's, Fr\'echet's and Chung's linearinequalities 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 withnon-decreasing $(k,l)$. The method of proof uses combinatorial identities, andreveals a multiplicative structure before taking expectation over samplepoints.
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机译：我们用二元二项式矩$ \ {S_ {i，j} \} $，$ 1获得$ P（S \ ge u，T \ ge v）$的Gumbel's，Fr''echet's和Chung's线性不等式。 $（S，T）$的联合分布的\ le i \ le k，1 \ le j \ le l $。在$ u = v = 1 $时，Gumbel和Fr'echet边界单调提高，且不减少$（k，l）$。证明方法使用组合身份，并在对样本点进行期望之前揭示乘法结构。
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