= Solution
A <credibility premium> estimates a risk's expected future claims by combining its own observed experience with a collective or prior mean. In its basic form $Z\overline X+(1-Z)\mu$, the <credibility factor> $Z\in[0,1]$ is the weight assigned to individual experience. Larger credibility places less weight on pooling with the population. The amounts below are pure expected-claim premiums; no additional expense or profit loading is specified.
Under <squared-error loss>, the posterior risk for estimating the unknown conditional mean $\theta$ by $a$ is
$$
\mathbb E[(\theta-a)^2\mid\boldsymbol X]=\operatorname{Var}(\theta\mid\boldsymbol X)+\left(a-\mathbb E[\theta\mid\boldsymbol X]\right)^2.
$$
Hence the <Bayes estimator under squared error loss> is the <posterior mean>. In the equal-year model, <conditional independence> makes the likelihood proportional to $\exp[-\sum_{j=1}^n(X_j-\theta)^2/(2\sigma_1^2)]$. Combining it with the normal prior and completing the square gives
$$
\theta\mid\boldsymbol X\sim N(m_n,V_n),\qquad V_n=\left(\frac1{\sigma_2^2}+\frac n{\sigma_1^2}\right)^{-1},\qquad m_n=V_n\left(\frac\mu{\sigma_2^2}+\frac{\sum_jX_j}{\sigma_1^2}\right).
$$
Because $\mathbb E[X_{n+1}\mid\theta]=\theta$, the required estimator is $m_n$; it also equals the predictive mean $\mathbb E[X_{n+1}\mid\boldsymbol X]$. The <normal-normal credibility> expression is
$$
\boxed{\widehat\theta=Z\overline X+(1-Z)\mu,\qquad Z=\frac{n\sigma_2^2}{\sigma_1^2+n\sigma_2^2}=\frac n{n+\sigma_1^2/\sigma_2^2}.}
$$
This is exact <Bayesian credibility>, not merely the best linear approximation to a possibly nonlinear <posterior mean>.
For unequal life counts use the usual exposure model: individual-life losses are conditionally independent with <variance> $v$, and the annual totals are conditionally independent given the same $\theta$. Then
$$
Y_j\mid\theta\ \mathrel{\dot\sim}\ N(m_j\theta,m_jv),\qquad X_j=Y_j/m_j\mid\theta\ \mathrel{\dot\sim}\ N(\theta,v/m_j).
$$
The dot denotes the stipulated normal approximation. The positive exposures $m_j$ are known. Define total past exposure and its experience mean by
$$
W=\sum_{j=1}^nm_j,\qquad\overline X_w=\frac{\sum_jm_jX_j}{W}=\frac{\sum_jY_j}{W}.
$$
The normal log-likelihood has quadratic term $-\sum_jm_j(X_j-\theta)^2/(2v)$. With the prior $N(\mu,\sigma^2)$, <normal-normal conjugacy with unequal exposures> gives <posterior variance> and mean
$$
V=\left(\frac1{\sigma^2}+\frac Wv\right)^{-1},\qquad m=V\left(\frac\mu{\sigma^2}+\frac{\sum_jY_j}{v}\right).
$$
Therefore the estimated pure premium per life for year $n+1$ has the <normal-normal credibility with unequal exposures> form
$$
\boxed{\widehat\pi_{n+1}=Z_W\overline X_w+(1-Z_W)\mu,\qquad Z_W=\frac{W\sigma^2}{v+W\sigma^2}=\frac W{W+v/\sigma^2}.}
$$
It is also an exact normal-model <Bühlmann–Straub credibility estimate>. The individual experience is exposure-weighted: the unweighted mean of the yearly per-life observations would generally discard the different precisions.
For positive $W,v,\sigma^2$, differentiation gives
$$
\frac{\partial Z_W}{\partial\sigma^2}=\frac{Wv}{(v+W\sigma^2)^2}>0,\qquad\frac{\partial Z_W}{\partial v}=-\frac{W\sigma^2}{(v+W\sigma^2)^2}<0.
$$
Thus \b[increasing the prior <variance> increases credibility], because greater between-risk heterogeneity makes the collective mean less representative of this risk. \b[Increasing the individual process <variance> decreases credibility], because a fixed amount of observed experience becomes noisier. The future exposure supplies no new observation, so it does not enter $Z_W$. It only scales the total premium:
$$
\boxed{\widehat\Pi_{n+1}=m_{n+1}\left[Z_W\frac{\sum_{j=1}^nY_j}{\sum_{j=1}^nm_j}+(1-Z_W)\mu\right].}
$$
The independent-life <variance> scaling is part of the usual exposure interpretation. Without it, a known <variance> $v$ for one life does not determine the <variance> of a total: a common pairwise conditional <covariance> $c$ would add $m_j(m_j-1)c$ to $m_jv$. Similarly, cross-year conditional dependence would change the joint likelihood. Those assumptions are implicit in applying the standard credibility model to the final paragraph.
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