For one Bernoulli variable, direct expansion gives $\operatorname{Var}F(X_1)=p(1-p)(F(1)-F(0))^2$. Applying the law of total variance successively to the coordinates gives the <variance tensorization>
$$
\boxed{\operatorname{Var}F(X)\leq p(1-p)\sum_{i=1}^n
\mathbb E\bigl(F(X)-F(X^i)\bigr)^2.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 2 / 9F / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ia/paper-2/9f/b/solution