Normalizing inputs (part 2)

The video contains a typo:
If X has mean $LaTeX: \mu$ $\mu$ and variance $LaTeX: \sigma^2$ $\sigma^2$ , then $LaTeX: Z=(X-\mu)/\sigma$ $Z=(X-\mu)/\sigma$ has mean LaTeX: 0 $0$ and variance LaTeX: 1 $1$ . That is, to normalize LaTeX: X $X$ we should divide by the standard deviation $LaTeX: \sigma$ $\sigma$ (not the variance $LaTeX: \sigma^2$ $\sigma^2$ as stated in the video).

We can verify the values of the mean and variance of LaTeX: Z $Z$ :

$LaTeX: \mathbb{E}[Z]=\mathbb{E}\left[\frac{X-\mu}{\sigma}\right] = \frac{\mu-\mu}{\sigma}=0$ $\mathbb{E}[Z]=\mathbb{E}\left[\frac{X-\mu}{\sigma}\right] = \frac{\mu-\mu}{\sigma}=0$
$LaTeX: \mathbb{E}[(Z-0)^2] = \mathbb{E}\left[\frac{(X-\mu)^2}{\sigma^2}\right]=1$ $\mathbb{E}[(Z-0)^2] = \mathbb{E}\left[\frac{(X-\mu)^2}{\sigma^2}\right]=1$ .