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If $c$ is a known positive constant and conditional on $\theta$, $X_1,...,X_n$ are independent RVs with pdf $f(x |\theta)=\theta c^\theta x^{-(\theta+1)}$. Given a prior distrbution of $Gamma(\alpha,\beta)$ I wanted to find out the posterior distribution of $\theta$.

From my working I ended up with the posterior distribution being proportional to $\theta^{n+\alpha-1}e^{-\beta\theta}c^{n\theta}m^{-\theta}$, where $m=\Pi_i x_i$ Just wanted to check I've done this correctly and also if this could be put in a closed form distribution?

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With your notation, $$f(\theta \mid \boldsymbol x) \propto f(\boldsymbol x \mid \theta) p(\theta) = \theta^n c^{n \theta} m^{-(\theta+1)} \theta^{\alpha-1} e^{-\beta \theta} \propto \theta^{n+\alpha-1} (c^n m e^{-\beta})^\theta.$$ This of course is proportional to a suitable gamma density, since $$c^n m e^{-\beta} = e^{-\beta + n \log c + \log m}.$$

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