 Inclusion indicator

An inclusion indicator records whether a specified population unit appears in a sample $S$:It is an indicator random variable, with expected value $\mathbb{E}{I}_i={\pi }_i$ and variance ${\pi }_i(1-{\pi }_i)$. Its randomness comes from sample selection; the population value attached to unit $i$ may be treated as fixed in a sampling calculation.^[ref]

Inclusion indicators turn a sum over a changing sample into a sum over fixed population labels:If every ${\pi }_i>0$, the same identity immediately provesThe inverse-inclusion-probability total is unbiased because $\mathbb{E}(I_i/{\pi }_i)=1$, not because the selected units must look typical in every realization.

Pair products connect the notation to joint inclusion probabilities: $\mathbb{E}(I_i{I}_j)={\pi }_{ij}$, so $\mathrm{Cov}(I_i,I_j)={\pi }_{ij}-{\pi }_i{\pi }_j$. For a fixed sample size $n$, ${\sum }_i{I}_i=n$ in every realization. Consequently,In simple random sampling with $N=4,n=2$, the four individual variances sum to $1$, while the six distinct pair covariances are each $-1/12$. Their doubled sum is $-1$, giving total variance zero. Treating the inclusion indicators as independent would incorrectly predict a random sample size. The notation makes that error easy to detect.