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Let us say we have a stochastic variable $X$ that is distributed according to some PDF $F$ that has parameters: $X\sim F(\alpha,\beta,...)$. How do one go about finding the PDF for $X$ if e.g. $\alpha \sim G(a,b,...)$ is also a stochastic variable?

For example, what is the pdf of $X\sim\mathcal{N}(\mu,\sigma^2)$ if $\sigma^2 \sim \chi^2(k)$?

(In the notation here $\mathcal{N}$ is the normal distribution and its PDF is then $N(x,\mu,\sigma^2)$, and $\chi^2$ is the chi-squared distribution and $F(x,k)$ is its PDF)

My first idea is to use conditional probability

$$ h(x|y) = \frac{f(x,y)}{f_y(y)} $$

where $f_y(y) = \int f(x,y) \mathrm{d}x$. First, using the example, we know that $h(x|\sigma^2) = N(x,\mu,\sigma^2)$ because when we fix $\sigma^2$ we will have just a normal PDF. Re-arranging the conditional probability

$$ h(x|\sigma^2) = \frac{f(x,\sigma^2)}{f_\sigma^2(\sigma^2)} \Leftrightarrow f_\sigma^2(\sigma^2)h(x|\sigma^2) = f(x,\sigma^2) $$

Because surly $X$ must be distributed according to the marginal of $f_x(x) = \int f(x,\sigma^2) \mathrm{d}\sigma^2$?

I am unsure of both this statement and if the following statement is correct: $f_\sigma^2(\sigma^2) = F(\sigma^2,k)$.

If these two statements are true, then the PDF of $X$ is

$$f_x(x) = \int F(\sigma^2,k) N(x,\mu,\sigma^2) \mathrm{d}\sigma^2$$

Daniel Korpi Kastinen
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    Your example is not okay since $\sigma^2$ does not take negative values. So it cannot have normal distribution. – drhab Dec 14 '17 at 15:37
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    The variance is positive and therefore cannot be distributed normally. I know this isn't the main point lf your question, but maybe pick another example? – Mathemagical Dec 14 '17 at 15:37
  • This is a very common setup in Bayesian framework. – BGM Dec 14 '17 at 15:51
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    @BGM What is "very common"? Almost surely nonnegative random variable with a normal distribution? – Did Dec 14 '17 at 16:10
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    If $X$ is conditionally normal $(\mu,\sigma^2)$ and $\sigma$ (not $\sigma^2$...) is normal $(0,a^2)$ then $X=\mu+aYZ$ where $(Y,Z)$ is i.i.d. standard normal. The distribution of $YZ$ is (definitely not normal but) described there. – Did Dec 14 '17 at 16:16
  • Ops, I didn't really have time to think through my example (since my actual problem is quite complicated I did not want to use that as an example), sorry for the inconvenience. I changed the example, and I also have a updated idea about how to think about it. – Daniel Korpi Kastinen Dec 14 '17 at 17:28
  • @Did I am looking for a more general approach here that will work even when one cannot extract parameters of the distributions from the stochastic variable, as one can easily do with normal distributions. Consider the non-central chi-squared distribution, here there is no closed form expression for PDF (it uses modified Bessel functions), if for example the non-centrality parameter was a stochastic variable one would have to use a more general derivation and then evaluate the resulting function numerically. – Daniel Korpi Kastinen Dec 14 '17 at 17:55

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