  Given a set of vertices $V$ with $|V| = n$, a weight vector $w \in
(\mathbb{R}^+ \cup \{ 0 \})^{\binom{V}{2}}$, and a probability vector $x \in
[0, 1]^{\binom{V}{2}}$ in the matching polytope, we study the quantity
$\frac{E_{G}[ \nu_w(G)]}{\sum_{(u, v) \in \binom{V}{2}} w_{u, v} x_{u, v}}$
where $G$ is a random graph where each edge $e$ with weight $w_e$ appears with
probability $x_e$ independently, and let $\nu_w(G)$ denotes the weight of the
maximum matching of $G$. This quantity is closely related to correlation gap
and contention resolution schemes, which are important tools in the design of
approximation algorithms, algorithmic game theory, and stochastic optimization.
  We provide lower bounds for the above quantity for general and bipartite
graphs, and for weighted and unweighted settings. he best known upper bound is
$0.54$ by Karp and Sipser, and the best lower bound is $0.4$. We show that it
is at least $0.47$ for unweighted bipartite graphs, at least $0.45$ for
weighted bipartite graphs, and at lea st $0.43$ for weighted general graphs. To
achieve our results, we construct local distribution schemes on the dual which
may be of independent interest.
