Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 2 / ii / Solution

For discrete $X$,Perturbing the marginal mass $p(x)$ contributes $\pi (X)\mu (X)-\beta$. Perturbing $\pi (x)=\mathbb{E}[A\mid X=x]$ contributesTheir sum is $A\mu (X)-\beta$. Finally, the influence function ofisMultiplication by the derivative $p(x)\pi (x)$ of $\beta$ with respect to $\mu (x)$ and summation over $x$ givesAdding the three contributions yields the claimed mean-zero influence curve