  We derive general properties of the finite-size scaling of probability
density functions and show that when the apparent exponent \tautilde of a
probability density is less than 1, the associated finite-size scaling ansatz
has a scaling exponent \tau equal to 1, provided that the fraction of events in
the universal scaling part of the probability density function is non-vanishing
in the thermodynamic limit. We find the general result that \tau>=1 and
\tau>=\tautilde. Moreover, we show that if the scaling function G(x) approaches
a non-zero constant for small arguments, \lim_{x-> 0} G(x) > 0, then
\tau=\tautilde. However, if the scaling function vanishes for small arguments,
\lim_{x-> 0} G(x) = 0, then \tau=1, again assuming a non-vanishing fraction of
universal events. Finally, we apply the formalism developed to examples from
the literature, including some where misunderstandings of the theory of scaling
have led to erroneous conclusions.
