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Date: 18-3-2021
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Given a random variable with continuous and strictly monotonic probability density function , a quantile function assigns to each probability attained by the value for which . Symbolically,
Defining quantile functions for discrete rather than continuous distributions requires a bit more work since the discrete nature of such a distribution means that there may be gaps between values in the domain of the distribution function and/or "plateaus" in its range. Therefore, one often defines the associated quantile function to be
where denotes the range of .
REFERENCES:
Shaw, W. "Refinement of the Normal Quantile: A Benchmark Normal Quantile Based on Recursion, and an Appraisal of the Beasley-Springer-Moro, Acklam, and Wichura (AS241) Methods." 2007. https://www.mth.kcl.ac.uk/~shaww/web_page/papers/NormalQuantile1.pdf.
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