 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 218

Conditional on $X_1,\ldots,X_n$, the $L$ selected class indicators are independent Bernoulli variables. Therefore
$$
e_1(x)=\frac1L\sum_{\ell=1}^Lp_1(X_{(\ell)})
$$
and
$$
\mathbb E\left[(\widehat p_1(x)-e_1(x))^2\mid X_1,\ldots,X_n\right]
=\frac1{L^2}\sum_{\ell=1}^Lp_1(X_{(\ell)})(1-p_1(X_{(\ell)}))
\leq\frac1{4L}\leq\frac1L.
$$
Taking expectations proves the claim.

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 2 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 2 / b / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 2 / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)