Distribution arising from randomly distributed mean and variance

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SUMMARY

The discussion centers on deriving the unconditional distribution of a random variable X, which follows a normal distribution X ~ N(μ, σ²), where both the mean μ and variance σ² are also normally distributed. Specifically, μ ~ N(μμ, σμ²) and σ ~ N(μσ, σσ²). The participants emphasize using Bayes' theorem to obtain the resulting distribution of X in terms of the parameters μμ, σμ, μσ, and σσ. The conclusion highlights the necessity of integrating over the distributions of μ and σ² to achieve the desired result.

PREREQUISITES
  • Understanding of normal distributions, specifically N(μ, σ²).
  • Familiarity with Bayesian statistics and Bayes' theorem.
  • Knowledge of conditional and unconditional distributions.
  • Basic calculus, particularly integration techniques.
NEXT STEPS
  • Study the application of Bayes' theorem in continuous distributions.
  • Learn about the properties of normal distributions and their transformations.
  • Explore the concept of marginalization in probability theory.
  • Investigate advanced topics in Bayesian inference, including hierarchical models.
USEFUL FOR

Statisticians, data scientists, and researchers involved in probabilistic modeling and Bayesian analysis will benefit from this discussion.

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forgive for my ignorance, but i have a practical problem that i don't 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]?
 
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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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