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

IGM Constraints from the SDSS-III/BOSS DR9 Ly-alpha Forest Flux Probability Distribution Function