Quantile function

In probability and statistics, the quantile function is a function ${\displaystyle Q:[0,1]\mapsto \mathbb {R} }$ which maps some probability ${\displaystyle x\in [0,1]}$ of a random variable ${\displaystyle v}$ to the value of the variable ${\displaystyle y}$ such that ${\displaystyle P(v\leq y)=x}$ according to its probability distribution. In other words, the function returns the value of the variable below which the specified cumulative probability is contained. For example, if the distribution is a standard normal distribution then ${\displaystyle Q(0.5)}$ will return 0 as 0.5 of the probability mass is contained below 0.

The quantile function is also called the percentile function (after the percentile), percent-point function, inverse cumulative distribution function (after the cumulative distribution function or c.d.f.) or inverse distribution function.