The standard normal distribution can be transformed into any general normal distribution via a location-scale transformation (X = μ + σZ), where μ shifts the distribution's location (its mean) and σ rescales its spread (its standard deviation); the inverse operation, standardization (Z = (X − μ)/σ), converts any normal random variable back to standard form and underlies derivation of the general normal's density via the chain rule applied to the standard normal's CDF. This sits within the broader theory of moments and variance in probability: variance is invariant under location shifts but scales quadratically under multiplicative rescaling, is not linear (in general, Var(X+Y) ≠ Var(X)+Var(Y) except under independence), and the Law of the Unconscious Statistician (LOTUS) permits computing expectations of transformed random variables directly from the original variable's density or mass function without deriving the transformed variable's distribution. These principles belong to probability theory's treatment of distribution families, transformations, and moment computation.
The standard normal distribution can be transformed into any general normal distribution via a location-scale transformation (X = μ + σZ), where μ shifts the distribution's location (its mean) and σ …