The sparsity function, also called the quantile density function, is the derivative of a quantile function 
,
 |
where 
and the derivative exists (Jones 1992, pp. 721-722;
NIST 1996, p. 8-1). If the distribution
function 
has a continuous probability density function 
that is positive
at 
,
then
 |
This follows by differentiating 
. The reciprocal-density identity does not apply to
a probability mass function of a discrete
distribution.
