# What is the distribution of standard normal CDF($x$) with normal prior on $x$?

Consider $\theta \sim N(\mu,\sigma^2)$. And let $\Phi(\cdot)$ denote the CDF of standard normal distribution.

Then what is the distribution of $\Phi(\theta)$?

## 1 Answer

a little suggestion:

$\theta \sim N(\mu,\sigma^2)$ means $\frac{\theta - \mu}{\sigma} \sim N(0,1)$.

You also know that $\Phi(x) = \frac{1}{\sqrt 2 \pi} \int_{- \infty}^x (e^{-x^2/2} dx)$, where $x \sim N(0,1)$

You can try to go on from here