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Just a note
Aside from the probability density function shown in elementary stats textbooks, the normal distribution can be defined by its cumulative (probability) distribution function, the inverse (quantile) distribution function and the cumulant function  and their calculus equivalents.
In addition, there are formulae which relate its characteristic function to moments about the origin  and an assortment of power series, asymptotic, continued fraction expansions, polynomial, and rational approximation functions.
