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 
.
