We investigate strong-coupling properties of a two-dimensional ultracold Fermi gas in the normal state. Including pairing fluctuations within the framework of a $T$-matrix approximation, we calculate the distribution function $n({boldsymbol Q})$ of Cooper pairs in terms of the center of mass momentum ${boldsymbol Q}$. In the strong-coupling regime, $n({boldsymbol Q}=0)$ is shown to exhibit a remarkable increase with decreasing the temperature in the low temperature region, which agrees well with the recent experiment on a two-dimensional $^6$Li Fermi gas [M. G. Ries, {it et. al.}, Phys. Rev. Lett. {bf 114}, 230401 (2015)]. Our result indicates that the observed remarkable increase of the number of Cooper pairs with zero center of mass momentum can be explained without assuming the Berezinskii-Kosterlitz-Thouless (BKT) transition, when one properly includes pairing fluctuations that are enhanced by the low-dimensionality of the system. Since the BKT transition is a crucial topic in two-dimensional Fermi systems, our results would be useful for the study toward the realization of this quasi-long-range order in an ultracold Fermi gas.
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