standardization
Distribution standardization is the statistical process of transforming data to have a mean of $0$ and a standard deviation of $1$,
often called Z-score normalization.
$$
Z = \frac{X - \mu}{\sigma}
$$
It converts different normal distributions into a single, standard normal distribution $Z \sim \mathcal{N}(0, 1)$
, enabling direct comparison of datasets, simplifying probability calculations, and optimizing machine learning algorithms.
Based on the properties of the expected value and variance,
$E(aX + b) = aE(X) + b $ and $Var(aX + b) = a^2Var(x)$, we can derive the following:
$$
\begin{aligned}
E(Z) &= E(\frac{1}{\sigma}X - \frac{1}{\sigma}\mu)
\,\\
\,\\
&= \frac{1}{\sigma}E(X) - \frac{1}{\sigma}\mu
\,\\
\,\\
&= \frac{1}{\sigma}\mu - \frac{1}{\sigma}\mu
\,\\
\,\\
&= 0
\,\\
\,\\
Var(Z) &= Var(\frac{1}{\sigma}X - \frac{1}{\sigma}\mu)
\,\\
\,\\
&= \frac{1}{\sigma^{\small{2}}}Var(X)
\,\\
\,\\
&= \frac{1}{\sigma^{\small{2}}}\sigma^{2}
\,\\
\,\\
&= 1
\end{aligned}
$$