Distribution arising from randomly distributed mean and variance

[omg!]

forgive for my ignorance, but i have a practical problem that i don't know how to approach:

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

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

 

[EnumaElish]

X's distribution is conditional on \mu and \sigma^2. 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.