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Just a note
Notice that, because x is in effect a quantile of population X, this distribution function is sometimes described as a quantile distribution function.
Also, following standard mathematical notation, the inverse function is indicated using the same symbol, followed by a 1 superscript. As with the cumulative distribution function, the inverse function (N^{1}) requires three values to calculate a quantile corresponding to P: P, μ and σ  whereas the 'standard' normal distribution function (Φ^{1}) only needs one value, P.
