From the definition of the variance:
$$\sigma^2 = \frac{1}{N-1}\sum_{i=0}^{N-1}(x_i-\mu)^2 \tag{1}$$
and mean:
$$\mu = \frac{1}{N}\sum^{N-1}_{i=0}x_i \tag{2}$$
how is it possible to derive the variance for running statistics, which is:
$$\sigma^2 = \frac{1}{N-1}\left[\sum^{N-1}_{i=0}x_i^2-\frac{1}{N}\left(\sum_{i=0}^{N-1}x_i \right)^2\right] \tag{3}$$
why the term $2x_i\mu$ is not considered?
Starting from $(1)$, I get:
$$\sigma^2 = \frac{1}{N-1}\left[\sum^{N-1}_{i=0}x_i^2-\sum_{i=0}^{N-1}2x_i \mu\right]+\frac{\mu^2}{N-1} $$