  The analysis of the Lyman-alpha forest of absorption lines in quasar spectra
has emerged as a potentially powerful technique to constrain the linear matter
power spectrum. In most previous work, the amplitude of the ionizing background
was fixed by calibrating simulations to match the observed mean transmitted
flux in the Lyman-alpha forest. This procedure is undesirable in principle as
it requires the estimation of the unabsorbed quasar continuum level, a
difficult undertaking subject to various sources of systematic error and bias.
We suggest an alternative approach based on measuring the one-point probability
distribution function (pdf) of the fluctuations in the flux about the mean,
relative to the mean, i.e. the pdf of delta_f = (f - <f>)/<f>. This statistic,
while sensitive to the amplitude of the ionizing background, has the virtue
that its measurement does not require an estimate of the unabsorbed continuum
level. We present a measurement of the pdf of delta_f from seven Keck HIRES
spectra, spanning a redshift range of z = 2.2 - 4.4. To illustrate that our
method is useful, we compare our measurements of the pdf of delta_f, and
measurements of the flux power spectrum from Croft et al. (2002) at z = 2.72,
with cosmological simulations. From this comparison, we obtain constraints on
the mean transmission in the Lyman-alpha forest, the slope of the
temperature-density relation, as well as the amplitude and slope of the mass
power spectrum. Our methodology will be useful for obtaining more precise
constraints with larger data samples from the Sloan Digital Sky Survey (SDSS).
