Covariance of draws without replacement ($\mathrm{Cov}(Z_1,Z_2)=-{\sigma }_N^2/(N-1)$)

Draw two distinct positions uniformly from a fixed population $z_1,\dots ,z_N$ 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 $\mu$, whileSubtracting 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.