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Concept# Quantile function

Summary

In probability and statistics, the quantile function outputs the value of a random variable such that its probability is less than or equal to an input probability value. Intuitively, the quantile function associates with a range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the percentile function (after the percentile), percent-point function or inverse cumulative distribution function (after the cumulative distribution function).
With reference to a continuous and strictly monotonic cumulative distribution function of a random variable X, the quantile function maps its input p to a threshold value x so that the probability of X being less or equal than x is p. In terms of the distribution function F, the quantile function Q returns the value x such that
which can be written as inverse of the c.d.f.
In the general case of distribution functions that are not strictly monotonic and therefore do not permit an inverse c.d.f., the quantile is a (potentially) set valued functional of a distribution function F, given by the interval
It is often standard to choose the lowest value, which can equivalently be written as (using right-continuity of F)
Here we capture the fact that the quantile function returns the minimum value of x from amongst all those values whose c.d.f value exceeds p, which is equivalent to the previous probability statement in the special case that the distribution is continuous. Note that the infimum function can be replaced by the minimum function, since the distribution function is right-continuous and weakly monotonically increasing.
The quantile is the unique function satisfying the Galois inequalities
if and only if
If the function F is continuous and strictly monotonically increasing, then the inequalities can be replaced by equalities, and we have:
In general, even though the distribution function F may fail to possess a left or right inverse, the quantile function Q behaves as an "almost sure left inverse" for the distribution function, in the sense that
almost surely.

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