= Covariance of draws without replacement
{title2=$\operatorname{Cov}(Z_1,Z_2)=-\sigma_N^2/(N-1)$}
Draw two distinct positions uniformly from a fixed population $z_1,\ldots,z_N$ with $N>1$. Let $\mu=N^{-1}\sum_i z_i$ and $\sigma_N^2=N^{-1}\sum_i(z_i-\mu)^2$. Each draw has <expected value> $\mu$, while
$$
\mathbb E[Z_1Z_2]=\frac{(\sum_i z_i)^2-\sum_i z_i^2}{N(N-1)}=\mu^2-\frac{\sigma_N^2}{N-1}.
$$
Subtracting the product of the <expected values> proves the <covariance> formula and explains the <finite population correction>. If the sampled values are coin biases and the actual tosses are <conditionally independent> given those biases, the <law of total covariance> gives the same <covariance> for their head <indicator random variables>.
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