For distinct <Type Ia supernova>[sibling supernovae] $s$ and $s'$ in the same galaxy, the shared fluctuation is the only common random term. Independence and <linearity of covariance> therefore give
$$
\operatorname{Cov}(M_s,M_{s'})=\sigma_G^2,
\qquad
\operatorname{Var}(M_s)=\sigma_G^2+\sigma_I^2
\equiv\sigma_{\rm tot}^2.
$$
Their <correlation coefficient> is
$$
\operatorname{Corr}(M_s,M_{s'})
=\frac{\sigma_G^2}{\sigma_G^2+\sigma_I^2}.
$$
This is the <intraclass correlation coefficient> of the galaxy-level <random intercept> model.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 219 / 1 / a / Solution

{title2=ICC}
{wiki}

The intraclass correlation coefficient measures the correlation between observations in the same group. In a random-intercept model with group variance $\sigma_G^2$ and individual variance $\sigma_I^2$, it is $\sigma_G^2/(\sigma_G^2+\sigma_I^2)$.


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