I have a random variable $X$ that is drawn from the pdf $$ f(x; \mu, \sigma, \sigma_{\mu}, \sigma_{\sigma}) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{|\hat{\sigma}|\sqrt{2\pi }} \frac{1}{ \sigma_\mu\sqrt{2\pi}} \frac{1}{\sigma_{\sigma}\sqrt{2\pi }} e^{\frac{-(x-\hat{\mu})^2}{2 \hat{\sigma}^2}}e^{\frac{-(\hat{\mu}-\mu)^2}{2\sigma_{\mu}^2}} e^{\frac{-(\hat{\sigma}-\sigma)^2}{2\sigma_{\sigma}^2}} d \hat{\sigma} d\hat{\mu}, $$ (with a proper normalization constant).
In essence, I first draw $\hat{\mu}$ from $\mathcal{N}(\mu, \sigma_{\mu})$ and draw $\hat{\sigma}$ from $\mathcal{N}(\sigma, \sigma_{\sigma})$, and then I draw $X$ from the distribution $\mathcal{N}(\hat{\mu},|\hat{\sigma}|)$.
In general I have not seen much about the problem of random variables drawn from random distributions, and have a number of basic questions about this situation:
Is the pdf given above the right function to describe the situation under it?
Is this pdf a known distribution, or can it be manipulated into a known form?
What are the moments of this function?
Is there any way to find/estimate the values $\sigma_{\mu}$ and $\sigma_{\sigma}$?
Really any information about this problem would be great.
