Write $k=2r+1$ and $m=n/k$. The <central limit theorem> gives jointly
$$
\frac{\sqrt m(\overline X_j-\mu)}\sigma
\Longrightarrow Z_j,
$$
where the $Z_j$ are independent standard normal variables. Hence
$$
\sqrt n(\widehat\mu_{\mathrm{MOM}}-\mu)
\Longrightarrow\sigma\sqrt k,Z_{(r+1)}.
$$
Its limiting cumulative distribution function is
$$
\sum_{j=r+1}^k\binom kj
\Phi\left(\frac{x}{\sigma\sqrt k}\right)^j
\left[1-\Phi\left(\frac{x}{\sigma\sqrt k}\right)\right]^{k-j}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 223 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-223/3/c/solution