  For a pair $(M, I)$, where $M$ is finitely generated graded module over a
standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with
$\ell(R/I) < \infty$, we introduce a new invariant $HKd(M, I)$ called the {\em
Hilbert-Kunz density function}. In Theorem 1.1, we relate this to the
Hilbert-Kunz multiplicity $e_{HK}(M,I)$ by an integral formula. We prove that
the Hilbert-Kunz density function is additive. Moreover it satisfies a
multiplicative formula for a Segre product of rings. This gives a formula for
$e_{HK}$ of the Segre product of rings in terms of the HKd of the rings
involved. As a corollary, $e_{HK}$ of the Segre product of any finite number of
Projective curves is a rational number. As an another application we see that
$e_{HK}(R, {\bf m}^k) - e(R, {\bf m}^k)/d!$ grows at least as a fixed positive
multiple of $k^{d-1}$ as $k\to \infty$.
