= Normal-normal credibility with unequal exposures
{title2=$Z=W\sigma^2/(v+W\sigma^2),\quad W=\sum_jm_j$}
For conditionally independent annual averages $X_j\mid\theta\sim N(\theta,v/m_j)$ and prior $\theta\sim N(\mu,\sigma^2)$, the <posterior mean> is $Z\overline X_w+(1-Z)\mu$, with $\overline X_w=\sum_jm_jX_j/W$. Completing the quadratic <likelihood> gives posterior precision $\sigma^{-2}+W/v$. This is an exact <Bayesian credibility> version of the <Bühlmann–Straub credibility estimate>. Increasing the between-risk prior variance increases the experience weight; increasing process variance decreases it. The next total pure premium is $m_{n+1}$ times the estimated per-life mean.
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