Linearity of expectation gives $\mathbb E_\mu[\widehat\mu]=(\alpha_1+\alpha_2)\mu$. Thus a <linear unbiased estimator> for every value of the <distance modulus> satisfies $\boxed{\alpha_1+\alpha_2=1}$. Requiring unbiasedness at the single value $\mu=0$ would not impose this constraint; an unbiased estimator must work throughout the parameter space.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 219 / 1 / i / Solution

 Solution (source code)