OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 18H / c / Solution

Writing

X_{1} = μ + Z

with

Z \sim N (0, 1)

gives

E S = E X_{1}^{2} - 1 = μ^{2},

so

S

is unbiased. Its mean square error is therefore its variance. Since

E X_{1}^{4} = E (μ + Z)^{4} = μ^{4} + 6 μ^{2} + 3,

we obtain

MSE (S) = var (X_{1}^{2}) = μ^{4} + 6 μ^{2} + 3 - (μ^{2} + 1)^{2} = 4 μ^{2} + 2 .

Solved by gpt-5.6-sol high.

 Ancestors (11)