A Multi-epoch X-ray Study of the Spiral Galaxy NGC 7331

X-ray point sources in galaxies are dominated by X-ray binaries (XRBs) that are variables or transients and whether their variability would alter the X-ray luminosity functions (XLF) is still in debate. Here we report on NGC 7331 as an example to test this with 7 Chandra observations. Their detection limit is 7$\times10^{37}$ erg s$^{-1}$ in the 0.3 -- 8.0 keV energy range by assuming a power-law spectral model with a photon index of 1.7. We detected 55 X-ray sources. Thirteen of them are variables for which 3 of them are transients and some of the sources possess a bimodal luminosity-hardness ratio feature, which is often observed among X-ray binaries. Nine more ultra-luminous X-ray sources are found comparing to previous studies and 8 are likely to be low-mass or high-mass XRBs. Twenty-one optical counterpart candidates are found based on the Hubble Space Telescope images, but we cannot rule out the possibility of positional coincidence. The spectral analysis of SN 2014C shows a trend of increasing soft photons and decreasing hydrogen column densities as its outer shell expands. We fit the 7 incompleteness-corrected XLFs to both a power-law (PL) and a power-law with an exponential cut-off (PLC) model using Bayesian method, which is the first time used in XLF fitting. The hierarchical PLC model can describe the XLF of NGC 7331 best with a slope of $\sim$ 0.5 and a luminosity cut-off around 8$\times$10$^{38}$ erg s$^{-1}$. This study proves that multi-epoch observations decrease the deviation due to the variable luminous sources in XLFs.