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I have a gamma distributed random variable $X$, with its mean $\mu$ distributed as some other function $$ X \sim \text{Gamma}(\mu,k)\\ \mu \sim P(\theta) $$

What is the distribution $P(\theta)$ such that when I marginalize out $\mu$, $X$ is still distributed as a gamma distribution, perhaps with updated parameters?

$$ p(x|k,\theta)=\int_0^{\infty} d\mu\, p(x|\mu,k)p(\mu|\theta) $$ i.e. $p(x|k,\theta)=\frac{x^{\kappa-1}e^{- x\kappa/\phi}}{(\phi/\kappa)^{\kappa}\Gamma(\kappa)}$ where $\phi,\kappa$ are functions of $k,\theta$

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  • $\begingroup$ You might require the self study tag. $\endgroup$ – Michael R. Chernick Oct 4 '19 at 0:06
  • $\begingroup$ Hint: the sum of Gamma distributions (with a common scale parameter) is a Gamma distribution. $\endgroup$ – whuber Oct 7 '19 at 21:27

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