Simplify $\mathbb{E}\left[ -\log \left( \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{z-\mu}{\sigma}\right)^{2}} \right) \right]$

Question

How to simplify the following expected value?

\begin{align} &= \mathbb{E}[ -\log f(x)] \\ &= \mathbb{E}\left[ -\log \left( \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{z-\mu}{\sigma}\right)^{2}} \right) \right]\\ &= ? \end{align}

Is the next step

$$\mathbb{E}\left[ -\log\frac{1}{\sigma \sqrt{2 \pi}}- \left( -\frac{1}{2}\left(\frac{z-\mu}{\sigma}\right)^{2} \right) \right] ?$$ and from there, how to reduce further to non-expectation form?

Answer 1

@Joitandr's approach is overkill, given what you've already done. By the linearity of expectation, the result is $\log(\sigma\sqrt{2\pi})+\frac{1}{2\sigma^2}\underbrace{\Bbb E[(z-\mu)^2]}_{\sigma^2}=\log(\sigma\sqrt{2e\pi})$.