  The probability density function of accretion disc luminosity fluctuations at
high observed energies (i.e., energies larger than the peak temperature scale
of the disc) is derived, under the assumption that the temperature fluctuations
are log-normally distributed. Thin disc theory is used throughout. While
log-normal temperature fluctuations would imply that the disc's bolometric
luminosity is also log-normal, the observed Wien-like luminosity behaves very
differently. For example, in contrast to a log-normal distribution, the
standard deviation of the derived distribution is not linearly proportional to
its mean. This means that these systems do not follow a linear rms-flux
relationship. Instead they exhibit very high intrinsic variance, and undergo
what amounts to a phase transition, in which the mode of the distribution (in
the statistical sense) ceases to exist, even for physically reasonable values
of the underlying temperature variance. The moments of this distribution are
derived using asymptotic expansion techniques. A result that is important for
interpreting observations is that the theory predicts that the fractional
variability of these disc systems should increase as the observed frequency is
increased. The derived distribution will be of practical utility in
quantitatively understanding the variability of disc systems observed at
energies above their peak temperature scale, including X-ray observations of
tidal disruption events.
