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· by Xianbin
\(\textbf{Lemma 1}\) (Second Moment Method). If \(X\) is a non-negative integer random variable, then we have
\[P(X=0) \leq \frac{\text{Var} X}{(EX)^2} = \frac{E[X^2]}{(EX)^2}-1\]
\(\textbf{Lemma 2}\) (Strong Second Moment Method). If \(X\) is a non-negative integer random variable, then we have
\[P(X=0) \leq \frac{\text{Var} X}{EX^2} = 1-\frac{(EX)^2}{E[X^2]}\]