I just wonder if pdf of Normal distribution with mean and variance which are normally distributed can be expressed in analytic formula, i.e.
$ \mathcal{N}(\mu, \sigma^2) $ where $ \mu $ ~ $ \mathcal{N}(m, d^2)$ and $\sigma$ ~ $Gamma(k, \theta)$
Let's say that I pick a random variable as time goes(t=1,2,.....,n). When I pick a random variable from $\mathcal{N}(\mu, \sigma^2)$ as time goes, each time the distribution is changed rather than fixed.
So, can $ \mathcal{N}(\mu, \sigma^2) $ where $ \mu $ ~ $ \mathcal{N}(m, d^2)$ and $\sigma$ ~ $Gamma(k, \theta)$ be expressed as analytic expression?