Distribution arising from randomly distributed mean and variance

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Main Question or Discussion Point

forgive for my ignorance, but i have a practical problem that i dont know how to approach:

[tex]X\sim\mathcal{N}(\mu,\sigma^2)[/tex]
where [tex]\mu\sim\mathcal{N}(\mu_{\mu},\sigma_{\mu}^2)[/tex]
and [tex]\sigma\sim\mathcal{N}(\mu_{\sigma},\sigma_{\sigma}^2)[/tex]

what is the resulting distribution of [tex]X[/tex], in terms of [tex]\mu_{\mu},\sigma_{\mu},\mu_{\sigma},\sigma_{\sigma}[/tex]?
 

Answers and Replies

  • #2
EnumaElish
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X's distribution is conditional on [itex]\mu[/itex] and [itex]\sigma^2[/itex]. Use Bayes' theorem (continuous version of Eq. 7 in http://mathworld.wolfram.com/BayesTheorem.html where integral replaces summation) to derive the unconditional distribution of X.
 

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