Given a random variable with continuous and strictly monotonic probability density function , a quantile function assigns to each probability attained by the value for which . Symbolically,
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 to be
where denotes the range of .