# Can we illustrate pdf of Normal distribution with mean and variance following Normal by analytic expression?

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?

• Indeed $\sigma^2$ can't be a Normal, since $\sigma^2>0$ and support of any normal is $(-\infty,\infty)$ – sinbadh Jan 19 '16 at 2:51
• Sorry. I was confused. I modified the error. Thank you. – qyong Jan 19 '16 at 3:13
• Seems to me similar to a Bayesian problem obtaining normal posterior from normal prior and normal data. – BruceET Jan 19 '16 at 4:04