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I am struggling with the following problem. I assume that there is a single claim number $X$ with corresponding heterogeneity parameter $\theta>0$.

I assume that $X$ given $\theta$ is Poisson distributed with parameter $\theta$, where $\theta$ has a continuous density $f_{\theta}$ on $(0,\infty)$ and I have derived that $\mathbb{E}[X|\theta]=\theta$.

I want to determine the conditional density $f_{\theta}(y|X=k)$ of $\theta$ given $X$ and I think that can be applied to calculate the Bayes estimator $m_k=\mathbb{E}[\theta|X=k]$.

Furthermore, I want to show that it holds that,

$$ m_k=(k+1)\frac{P(X=k+1)}{P(X=k)} $$

and verify that for $n\geq 1m$ it holds that,

$$ \mathbb{E}[\theta^n|X=k]=\prod_{j=1}^{n-1}m_{k+j} $$

Thanks in advance-

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  • $\begingroup$ What is $P$ in $P(X=k+1)$ and $P(X=k)$? Under which distribution is this calculated? Isn't $X$ observed already? $\endgroup$ Nov 10 '20 at 11:45
  • $\begingroup$ Under a Poisson distribution. $\endgroup$ Nov 10 '20 at 12:35

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