For binary $A$, conditional covariance satisfies
$$
\operatorname{Cov}(A,Y\mid X)
=\pi(X)\{1-\pi(X)\}
\left\{\mathbb E(Y\mid A=1,X)-\mathbb E(Y\mid A=0,X)\right\}.
$$
By <conditional exchangeability> and <consistency of potential outcomes>, the difference in braces is $\mathbb E[Y(1)-Y(0)\mid X]$. Consequently
$$
\beta
=\mathbb E\!\left[
\pi(X)\{1-\pi(X)\}\{Y(1)-Y(0)\}
\right],
$$
so the required weight is
$$
w(X)=\pi(X)\{1-\pi(X)\}.
$$
This <overlap weight> emphasizes covariate strata with treatment probabilities near one half and downweights strata near a violation of <positivity in causal inference>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 221 / 4 / i / Solution

By <consistency of potential outcomes> and <conditional exchangeability>,
$$
\mathbb E[Y(z)\mid X]
=\mathbb E[Y(z)\mid Z=z,X]
=\mathbb E[Y\mid Z=z,X].
$$
Taking conditional expectations given $X_1=x_1$ and subtracting the cases $z=1$ and $z=0$ proves the outcome-regression identification formula.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 221 / 4 / a / i / Solution

The raw difference is generally not a <causal effect>. Existing UTCs may differ systematically from non-UTCs in location, admissions, pupil composition, prior attainment, resources, and the local apprenticeship market. These are possible <confounders> of school type and outcome. The comparison also weights the 43 UTCs and roughly 3500 controls very differently from a clearly defined target population. Without <conditional exchangeability>, <positivity in causal inference>[positivity], and <consistency in causal inference>[consistency], the difference mixes the effect of UTC status with selection bias.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 1 / b / Solution

Conditional exchangeability requires $(Y(0),Y(1))\perp A\mid X$ for sufficient pretreatment covariates $X$.


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