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04 Existence proofs

$Pr(X\geq E[X])>0$, $Pr(X\leq E[X])>0$.

For sparse dependency graphs

Aka Lovasz local lemma.

For Dependency graph with $Pr(E_{i})<p, 4dp<1$: $Pr(\cap \bar{E_{i}}) > 0$.

Lovasz local lemma: general case: Dependency. graph G=(V,E), \ $x_{i} \in [0,1]$: $Pr(E_{i}) \leq x_{i} \prod_{(i,j) \in E}(1-x_{j})$: $Pr(\cap \bar{E_{i}})\geq \prod_{i}(1-x_{i}) > 0$

Threshold behavior

$X>0$: Second moment method: $Pr(X=0)\leq Pr((X-E[X])^{2} \geq (E[X])^{2})$. Conditional expectation inequality for $\sum$ indicators: \ $Pr(X>0) \geq \sum_{i=1}^{n}\frac{Pr(X_{i}=1)}{E[X|X_{i}=1]}$.

Explicit constructions

\tbc

Make Existence proofs

Design sample space, show $Pr(E)>0$, maybe modify to get final object. Use Expectation argument. Make non-negative RV X, use Chebyshev to bound Pr(X=0). Make dependency graph, use Lovasz local lemma.