= Solution
Conditional on $\lambda$, a <Poisson distribution> has both <conditional expectation> and <conditional variance> equal to $\lambda$. For the <uniform distribution> on $(1,3)$,
$$
m_0=\mathbb E\lambda=2,\qquad
a=\operatorname{Var}(\lambda)=\frac{(3-1)^2}{12}=\frac13,\qquad
v=\mathbb E\lambda=2.
$$
Thus $K=v/a=6$ and the one-year <credibility factor> is $Z_1=1/7$. With the observed count equal to one, the <Bühlmann credibility estimate> of the next year's conditional claim <expected value> is
$$
\boxed{\widehat m_1=\frac17\cdot1+\frac67\cdot2=\frac{13}{7}\approx1.85714.}
$$
The estimate gives relatively little weight to one year because the <expected process variance> is six times the <variance of hypothetical means>.
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