Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 2 / a / Solution

Write $X_{-i}$ for all coordinates except $X_i$, let $X_i^{\prime }$ be an independent random variable with the same distribution as $X_i$, and let $Z_i^{\prime }=f(X_1,\dots ,X_i^{\prime },\dots ,X_n)$. Three equivalent forms of the Efron–Stein inequality arefor arbitrary square-integrable $Z_i$ measurable with respect to $X_{-i}$, andThe first is the sharp choice within the second because conditional expectation is the least-squares projection. The first and third right sides are equal because two conditionally independent copies have expected squared difference twice their conditional variance.