  \textit{Extreme Ultraviolet Explorer} Deep Survey observations of cool stars
(spectral type F to M) have been used to investigate the distribution of
coronal flare rates in energy and its relation to activity indicators and
rotation parameters. Cumulative and differential flare rate distributions were
constructed and fitted with different methods. Power laws are found to
approximately describe the distributions. A trend toward flatter distributions
for later-type stars is suggested in our sample. Assuming that the power laws
continue below the detection limit, we have estimated that the superposition of
flares with radiated energies of about $10^{29}-10^{31}$ergs could explain the
observed radiative power loss of these coronae, while the detected flares are
contributing only $\approx 10$%. While the power-law index is not correlated
with rotation parameters (rotation period, projected rotational velocity,
Rossby number) and only marginally with the X-ray luminosity, the flare
occurrence rate is correlated with all of them. The occurrence rate of flares
with energies larger than $10^{32}$ergs is found to be proportional to the
average total stellar X-ray luminosity. Thus, energetic flares occur more often
in X-ray bright stars than in X-ray faint stars. The normalized occurrence rate
of flares with energies larger than $10^{32}$ergs increases with increasing
\lxlbol and stays constant for saturated stars. A similar saturation is found
below a critical Rossby number. The findings are discussed in terms of simple
statistical flare models in an attempt to explain the previously observed trend
for higher average coronal temperatures in more active stars. It is concluded
that flares can contribute a significant amount of energy to coronal heating in
active stars.
