Suppose we have a variable $\xi\sim N(\mu_2,\sigma_2)$ and another conditional variable $\eta | \xi \sim N(\mu_1,\sigma_1)$. Is $\eta$ normally distributed, and if so what is its mean and variance?
I think this has something to do with Bayes rule and the fact that the conditionals and marginals of multivariate normals are all normals, but I cannot prove it.
Note that the characteristic function of $\eta$ is \begin{align*} \varphi_\eta(t) &= \mathbb{E}e^{it\eta} \\ &= \mathbb{E}[\mathbb{E}[e^{it\eta}|\xi]] \\ &= e^{-\sigma_1^2t^2/2}\mathbb{E}[e^{it\mu_2(\xi)}] \end{align*} If $\mu_2(\xi) = a + b\xi$, then the result is the characteristic function for $N(a + b\mu_1, \sigma_1^2 + b^2\sigma_2^2)$. Otherwise, for example $\mu_2(\xi) = \xi^2$, then $\eta$ is something that is pretty chaotic; for general $\mu_2(\xi)$, $\eta$ is not always normal.