The Compound Parabolic Refractive X-ray Lenses (CPRXLs) are a kind of novel optical components, especially suitable for hard X-rays. Due to their parabolic profiles, they are free of spherical aberration and are genuine devices for sub-micrometer focusing and imaging. The properties of a rotationally symmetric parabolic X-ray lens are studied theoretically and numerically in this paper. Firstly, the exact focal length formula is derived by using matrix optics method and a criterion of the thin lens approximation is obtained. Secondly, according to the diffraction theory, intensity distribution near focus is analyzed, which can give the limited focal spot size. Finally, the theoretical formulas of effective aperture and the transmittance are deduced. In addition, the numerical results of the designed CPRXLs made of various materials are also shown.