  The Ly$\alpha$ forest transmission probability distribution function (PDF) is
an established probe of the intergalactic medium (IGM) astrophysics, especially
the temperature-density relationship of the IGM. We measure the transmission
PDF from 3393 Baryon Oscillations Spectroscopic Survey (BOSS) quasars from SDSS
Data Release 9, and compare with mock spectra that include careful modeling of
the noise, continuum, and astrophysical uncertainties. The BOSS transmission
PDFs, measured at $\langle z \rangle = [2.3,2.6,3.0]$, are compared with PDFs
created from mock spectra drawn from a suite of hydrodynamical simulations that
sample the IGM temperature-density relationship, $\gamma$, and temperature at
mean-density, $T_0$, where $T(\Delta) = T_0 \Delta^{\gamma-1}$. We find that a
significant population of partial Lyman-limit systems with a column-density
distribution slope of $\beta_\mathrm{pLLS} \sim -2$ are required to explain the
data at the low-transmission end of transmission PDF, while uncertainties in
the mean Ly$\alpha$ forest transmission affect the high-transmission end. After
modelling the LLSs and marginalizing over mean-transmission uncertainties, we
find that $\gamma=1.6$ best describes the data over our entire redshift range,
although constraints on $T_0$ are affected by systematic uncertainties. Within
our model framework, isothermal or inverted temperature-density relationships
($\gamma \leq 1$) are disfavored at a significance of over 4$\sigma$, although
this could be somewhat weakened by cosmological and astrophysical uncertainties
that we did not model.
