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Quantile Function


Given a random variable X with continuous and strictly monotonic probability density function f(X), a quantile function Q_f assigns to each probability p attained by f the value x for which Pr(X<=x)=p. Symbolically,

 Q_f(p)={x:Pr(X<=x)=p}.

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 Q_f to be

 Q_f(p)=inf{x in R(f):p<=f(x)},

where R(f) denotes the range of f.


See also

Continuous Distribution, Discrete Distribution, Distribution Function, Monotone Increasing, Monotonic Function, Probability, Probability Density Function, Quantile, Random Variable

This entry contributed by Christopher Stover

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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. http://www.mth.kcl.ac.uk/~shaww/web_page/papers/NormalQuantile1.pdf.

Cite this as:

Stover, Christopher. "Quantile Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/QuantileFunction.html

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