  Analytical expressions for the rates of formation and destruction of
gravitationally bound systems are derived assuming that they are originated
from primordial random-Gaussian density fluctuations. The resulting formulae
reproduce the time derivative of the Press-Schechter mass function in a certain
limit. Combining a theoretical model for the evolution of structures with the
formation rate, we can make various cosmological predictions which are to be
compared with observations. As an example to elucidate such applicability, we
evaluate the contribution of clusters of galaxies to the cosmic X-ray
background radiation. With the {\it COBE} normalization, we find that the
significant fraction of the observed soft X-ray background is accounted for by
clusters of galaxies in a cold dark matter universe with $\Omega_0 \sim 0.2$,
$\lambda_0 =1- \Omega_0$ and $h \sim 0.8$.
