# Poisson distributed claim in non life insurance mathematics

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,

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

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