 Solution

The structural parameters remain $m_0=2$, $v=2$ and $a=1/3$, so the credibility factor after $n$ years is $Z_n=n/(n+6)$. Writing $C_n={\sum }_{i=1}^n{x}_i$, the Bühlmann credibility estimate isThe target is $\mathbb{E}(X_{n+1}\mid \lambda )=\lambda$. It is also the best affine predictor of the next year's count, because its extra conditional noise has zero covariance with past observations.