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D. G. Horvitz coauthored the 1952 paper that generalized finite-population sampling without replacement to unequal selection probabilities. The joint work with D. J. Thompson provides the foundation for the Horvitz-Thompson estimator.^[ref]

The central idea is to compensate for unequal chances of inclusion. For fixed population values $y_i$ and positive inclusion probabilities ${\pi }_i$, write a total estimate asUsing an inclusion indicator $I_i$ gives $\mathbb{E}(I_i/{\pi }_i)=1$, so $\mathbb{E}\hat{Y}={\sum }_i{y}_i$. Unbiasedness follows from the design probabilities rather than from assuming that the sampled values are an independent realization of a population model.^[ref]

For an illustrative population $(4,6)$, select exactly one unit with probabilities $(1/4,3/4)$. The total estimate is $16$ or $8$, respectively, with expected value $(1/4)16+(3/4)8=10$, the true total. The less frequently selected unit receives the larger multiplier. This example explains how a sampling design can deliberately favor some units without forcing an unweighted analysis to be correct.

The Horvitz-Thompson estimator estimates a finite population total by expanding each observed unit's contribution by the reciprocal of its inclusion probability. D. G. Horvitz and D. J. Thompson introduced the general unequal-probability construction in 1952.^[ref]

Let $y_1,\dots ,y_N$ be fixed values attached to labelled population units. A sampling design selects a random subset $S$, with known inclusion probabilities ${\pi }_i=\mathbb{P}(i\in S)>0$. Thenwhere $I_i$ is the inclusion indicator. This is inverse probability weighting for a finite population.^[ref] Its expected value is immediatelyThe subscript $p$ emphasizes design-based inference: the fixed values are not assumed to follow a probability model; the sample-selection mechanism supplies the randomness. The inclusion indicators need not be independent. Dividing by known $N$ gives an unbiased estimator of the finite population mean.

For an original exact example, use values $(y_1,y_2,y_3)=(2,4,6)$ and choose samples $\{1,2\}$, $\{1,3\}$, $\{2,3\}$ with probabilities $1/2,1/3,1/6$, respectively. The inclusion probabilities are $(5/6,2/3,1/2)$. The corresponding estimates of the total are $8.4,14.4,18$, andSimply multiplying each sample mean by three instead gives totals $9,12,15$, with expected value $11$. Equal treatment of the observed values ignores the unequal chances of observing them.

The sampling variance depends on pair information. Put ${\pi }_{ij}=\mathbb{P}(i,j\in S)$, including ${\pi }_{ii}={\pi }_i$. Bilinearity of covariance givesFor the three-unit design it is $14.4$, found directly from $\frac{1}{2}(8.4-12{)}^2+\frac{1}{3}(14.4-12{)}^2+\frac{1}{6}(18-12{)}^2$. Thus an unbiased estimator need not give a nearly correct total in each sample.

With pairwise inclusion positivity, the same expansion device applied to pairs constructs the Horvitz-Thompson variance estimator:Its expected value equals the preceding population sum because ${\mathbb{E}}_p(I_i{I}_j/{\pi }_{ij})=1$. In the worked design its three possible values are $9.76,58.56,-60$, whose probability-weighted average is $14.4$. This particular unbiased estimator of variance can therefore be negative for an individual sample. Its expected value identity is a statement about repeated selection.

If ${\pi }_i=0$, an arbitrary unobserved $y_i$ cannot contribute to an unbiased total estimate. If a distinct pair has ${\pi }_{ij}=0$, its product cannot be recovered by the pair-expansion argument. Finally, dividing the weighted sum by the random sum of its inverse-probability weights changes the estimator into a ratio; the indicator proof above no longer establishes that this ratio is an unbiased estimator. The known population size and the actual sampling design are part of the result.

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.

A joint inclusion probability is the probability that two specified population units both belong to a random sample $S$:For the same unit twice, ${\pi }_{ii}={\pi }_i$, where ${\pi }_i$ is its inclusion probability. For distinct units, ${\pi }_{ij}$ is generally not ${\pi }_i{\pi }_j$: selection can make their inclusion indicators dependent. These probabilities describe the sampling procedure, rather than a relationship between the measured values of the two units.^[ref]

For simple random sampling of $n$ distinct labels from $N>1$ labels, counting subsets containing both $i$ and $j$ giveswith value zero if $n<2$. Since ${\pi }_i=n/N$,For $N=4,n=2$, each unit has inclusion probability $1/2$, but a particular pair has joint inclusion probability $1/6$. Conditional on including one member, the other is included with probability $(1/6)/(1/2)=1/3$, not $1/2$.

Equal individual inclusion probabilities do not determine these pair probabilities. A different procedure that chooses only $\{1,2\}$ or $\{3,4\}$, each with probability $1/2$, still includes each unit with probability $1/2$. Yet ${\pi }_{12}=1/2$ and ${\pi }_{13}=0$. Thus two procedures can agree on individual representation while producing different dependence and different precision for a population summary. The distinction motivates keeping pair information when calculating the variance of weighted survey estimators; the classical general treatment is due to D. G. Horvitz and D. J. Thompson.^[ref]

In finite-population sampling, a simple random sample of fixed size $n$ without replacement is a subset $S$ of $N$ labelled units drawn so that every size-$n$ subset has probabilityThe sample is the selected subset; simple random sampling is the procedure that generates it. This definition does not require the observed values to be distinct, only the selected population labels. The without-replacement convention is important because an independent repeated draw from a population is a different design.^[ref]

One construction randomly orders all $N$ labels and takes the first $n$. Each particular subset can occupy those positions in $n!(N-n)!$ permutations, out of $N!$, proving the probability above. Equivalently, draw successively and uniformly from the remaining labels, then discard their order.

For $N=5,n=2$, there are $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$ possible samples. A particular unit belongs to four of them, so its inclusion probability is $2/5$; a particular pair is one complete sample, so its joint inclusion probability is $1/10$. These are properties of the procedure before observing which subset was obtained.

A simple random sample need not resemble the whole population in every realization. For fixed values $(0,0,0,0,100)$, six of the ten size-two samples contain only zeros. Thus the sample mean is zero with probability $0.6$, although its expected value is the finite population mean $20$. The remaining four samples each have mean $50$, giving $0.6\times 0+0.4\times 50=20$. Unbiasedness concerns repeated selection, rather than a guarantee about any one sample. Coverage of the population labels and faithful implementation of the selection rule remain essential.