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 being less than or equal to this value is equal to the given probability. It is also called the percent point function or inverse cumulative distribution function.
Definition
With reference to a continuous and strictly monotonic distribution function, for example the cumulative distribution function
![\scriptstyle F_X\colon R \to [0,1]](http://assets.wn.com/wiki/en/8/1f/c75eb6823e1b364bf0f67-5274cb.png)
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.
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, then the infimum function can be replaced by the minimum function and
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