Quantile function
In probability and statistics, the quantile function specifies, for a given probability in the probability distribution of a random variable, the value at which the probability of the random variable is less than or equal to the given probability. It is also called the percent-point function or inverse cumulative distribution function.
With reference to a continuous and strictly monotonic distribution function, for example the cumulative distribution function of a random variable X, the quantile function Q returns a threshold value x below which random draws from the given c.d.f would fall p percent of the time.
![\scriptstyle F_{X}\colon R\to [0,1]](https://wikimedia.org/api/rest_v1/media/math/render/svg/5513857993bae964db86319ae52815a65e68f43a)
In terms of the distribution function F, the quantile function Q returns the value x such that
Another way to express the quantile function is
for a probability 0 < p < 1. 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.
If the function F is continuous and strictly monotonically increasing, then the infimum function can be replaced by the minimum function and
However, if F has flat regions or is discontinuous, then the inverse of F is not well-defined. In this case we need to use the more complicated formula above.
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