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Distribution arising from randomly distributed mean and variance

  1. May 16, 2010 #1
    forgive for my ignorance, but i have a practical problem that i dont know how to approach:

    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]?
  2. jcsd
  3. May 16, 2010 #2


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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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