= Second-moment bound for positivity
{title2=$\mathbb P(N>0)\geq(\mathbb EN)^2/\mathbb E N^2$}
For a nonnegative <random variable> $N$ with positive <expected value> and finite <second moment>, write $N=N\mathbf1_{\{N>0\}}$. The <Cauchy-Schwarz inequality> gives
$$
(\mathbb EN)^2=\left(\mathbb E[N\mathbf1_{\{N>0\}}]\right)^2\leq\mathbb E N^2\,\mathbb P(N>0),
$$
which proves the bound in <second moment method>. It is the zero-threshold case of the <Paley-Zygmund inequality>. Equality occurs when $N$ is a fixed positive value on the <event> $N>0$ and zero elsewhere, <almost surely>. For $N=10B$ with $B$ a <Bernoulli random variable> of parameter $1/10$, $(\mathbb EN)^2/\mathbb E N^2=1/10$ equals the exact positivity probability.
The <variance> identity $\mathbb E N^2=(\mathbb EN)^2+\operatorname{Var}(N)$ turns the lower bound into a statement about variance relative to the squared <mean>. A small ratio forces positivity to be likely. A positive <expected value> alone does not do so: keeping $\mathbb EN=1$ while taking $N=1/q$ with <probability> $q$ and zero otherwise makes the positivity probability as small as any chosen $q>0$.
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