Standardization of a normal random variable
= Standardization of a normal random variable
If $X\sim N(\mu,\sigma^2)$ with $\sigma>0$, then $Z=(X-\mu)/\sigma$ has the <standard normal distribution>. The <change-of-variables formula for a probability density> gives $f_Z(z)=\sigma f_X(\mu+\sigma z)=\phi(z)$, the <standard normal density>. Thus <standardization of a normal random variable> removes both parameters without changing the <probabilities> of corresponding <events>.