We have conducted a large-field simultaneous survey of $^{12}$CO, $^{13}$CO, and C$^{18}$O $J=1-0$ emission toward the Orion A giant molecular cloud (GMC) with a sky coverage of $sim$ 4.4 deg$^2$ using the PMO-13.7 m millimeter-wavelength telescope. We use the probability distribution function of the column density (N-PDF) to investigate the distribution of molecular hydrogen in the Orion A GMC. The H$_2$ column density, derived from the $^{13}$CO emission, of the GMC is dominated by log-normal distribution in the range from $sim$4$times10^{21}$ to $sim$1.5$times10^{23}$ cm$^{-2}$ with excesses both at the low-density and high-density ends. The excess of the low-density end is possibly caused by an extended and low-temperature ($sim$10 K) component with velocities in the range of 5$-$8 km s$^{-1}$. Compared with the northern sub-regions, the southern sub-regions of the Orion A GMC contain less gas with column density in $N_{H_2} > 1.25times 10^{22} rm{cm}^{-2}$. The dispersions of the N-PDFs of the sub-regions are found to correlate with the evolutionary stages of the clouds across the Orion A GMC. The structure hierarchy of Orion A GMC is explored with the DENDROGRAM algorithm, and it is found that the GMC is composed of two branches. All structures except one in the tree have virial parameters less than 2, indicating self-gravity is important on the spatial scales from $sim$0.3 to $sim$4 pc. Although power-laws and departures from log-normal distributions are found at the high-density end of N-PDFs of active star-forming regions, the N-PDFs of structures in the Orion A GMC are predominantly log-normal on scales from R$sim$0.4 to 4 pc.
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