To look for possible phenomenological connections between pulsars timing properties and emissions from pulsar wind nebulae and their pulsars, we studied the power-law component of the X-ray emissions from 35 pulsar wind nebulae which have a detected pulsar in X-rays. Our major results are in the following: (1) The power-law component of the X-ray luminosities, in the energy range from 0.5 keV to 8 keV, of the nebulae and of the pulsar both show a strong correlation with the pulsar spin-down power ($dot{E}$), consistent with earlier studies. However, equally significant correlations with the magnetic field strength at the light cylinder ($B_{rm lc}$) are also found. The similar significance level of the correlations with $dot{E}$ and with $B_{rm lc}$ suggests that not only $dot{E}$ but also $B_{rm lc}$ plays an important role in understanding these power-law emissions. (2) Thermal X-ray emissions are detected in 12 pulsars among the 35 samples. With derived temperature as one additional variable, we found that the photon indices of pulsars non-thermal X-ray power-law spectra can be well described by a linear function of $log P$, $logdot{P}$ and temperature logarithm $log T$. It indicates that the surface temperature of neutron stars plays an important role in determining the energy distribution of the radiating pair plasma in pulsars magnetospheres.
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