  Fermi-LAT LCR provide continuous and regularly-sampled gamma-ray light
curves, spanning about 14 years, for a large sample of blazars. The log-normal
flux distribution and linear RMS-Flux relation of the light curves for a few of
Fermi blazar have been examined in previous studies. However, the probability
that blazars exhibit log-normal flux distribution and linear RMS-Flux relation
in their gamma-ray light curves has not been systematically explored. In this
study, we comprehensively research on the distribution of gamma-ray flux and
the statistical characteristics on a large sample of 1414 variable blazars from
the Fermi-LAT LCR catalog, including 572 FSRQs, 477 BL Lacs, and 365 BCUs, and
statistically compare their flux distributions with normal and log-normal
distributions. The results indicate that the probability of not reject
log-normal is 42.05% for the large sample, and there is still 2.05% probability
of not reject normality, based on the joint of Kolmogorov-Smirnov, Shapiro-Wilk
and Normality tests. We further find that the probability that BL Lacs conforms
to the log-normal distribution is higher than that of FSRQs. Besides, after
removing sources with less than 200 data points from this large sample, a
sample of 549 blazars, which is still a large sample comparing to the previous
studies, was obtained. Basing on dividing the light curves into segments every
20 points (or 40 points, or one year), we fitted the linear RMS-Flux relation
of this three different sets, and found that the Pearson correlation
coefficients are all close to 1 of the most blazars. This result indicates a
strong linear correlation between the RMS and the flux of this 549 blazars. The
log-normal distribution and linear RMS-Flux relation indicate that the
variability of gamma-ray flux for most blazars is non-linear and multiplicative
process.
