Active mixers usually include an amplifier, either before or after the converting mixer. Passive mixers alone have a conversion loss, typically 6-10 dB. Active mixers can have a conversion gain due to amplifier gain.
To know mixer noise figure, in the passive mixers it equals to conversion loss, if both RF sidebands (above and below LO frequency) are converted to IF. If one sideband is suppressed by a RF filter, however, the noise figure is ~3 dB higher.
With active mixers, the noise figure depends on where the amplifier is located. If before the mixer, then amplifier noise figure determines the system noise figure. Otherwise, one must use the Friis' formula to include gains and losses and noise figures of individual components.