In recent years, there has been an increasing use of fractional differential equations due to their ability to effectively represent various engineering phenomena, including viscoelasticity, heat transport, non-local continuum, and others. These equations take into account certain effects that cannot be accurately predicted using classical differential equations.This paper provides a comprehensive analysis of the fractional compound motion, specifically focusing on the response of a one-term fractional differential equation that is excited by a Poissonian white noise process. The present study introduces a straightforward equation for the probability density function of fractional compound motion. The validity of this equation is subsequently confirmed by the execution of various numerical simulations. Furthermore, a comprehensive analysis is conducted on the self-similarity of fractional compound motion, demonstrating that the phenomenon can be regarded as self-similar in weak sense. This characteristic can be effectively employed to mitigate the loss of Markovianity in fractional differential equations.