standard normal distribution

A standard normal distribution is a normal distribution with a mean $\mu =0$ and unit standard deviation $\sigma =1$.

standard normal distribution CDF ($\Phi$)

For the standard normal distribution ($\mu =0,\sigma =1$), the CDF is often denoted as $\Phi (x)$.

For the interval $[-1,1]$, which represents 1 standard deviation from the mean:

$$P(-1\le X\le 1)=\Phi (1)-\Phi (-1)\approx 0.8413-0.1587=0.6826$$

This shows that approximately 68% of values drawn from a standard normal distribution lie within one standard deviation of the mean.

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Isotropic gaussians are rotation invariant around their mean. E.g. standard normal distribution: If $X\sim \mathcal{N}(0,I)$ and $Q$ is an orthogonal matrix, then $QX\sim \mathcal{N}(0,I)$.

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