Majorana fermions are their own antiparticles, which play an important role in fault-tolerant topological quantum computation. Recently, the search for Majorana fermions in condensed matter physics, is attracting a great deal of attention as quasiparticles emerge. In this paper we consider a specific model consisting of double quantum dots and a tunnel-coupled semiconductor nanowire on an s-wave superconductor, since the nanowire may support Majorana fermions under appropriate conditions. We study the electron transport through the double quantum dots by using the particle-number resolved master equation. We pay particular attention to the effects of Majorana's dynamics on the current fluctuation (shot noise). It is shown that the current and the shot noise measurement can be used to distinguish Majorana fermions from the usual resonant-tunneling levels. When there exist Majorana fermions coupling to the double quantum dots, a difference between the steady-state source and drain currents depends on the asymmetry of electron tunneling rates. The asymmetric behaviors of the currents can reveal the essential features of the Majorana fermion. Moreover, the dynamics of Majorana coherent oscillations between the semiconductor nanowire and the double quantum dots is revealed in the shot noise, via spectral dips together with a pronounced zero-frequency noise enhancement effect. We find, on the one hand, that the peak of the zero-frequency noise becomes a dip in the case of weak coupling between double quantum dots and the nanowire; on the other hand, for the strong coupling the dip of the zero-frequency noise becomes even further deep with side dips towards high frequency regimes. Furthermore, the dip of the zero-frequency noise disappears and a zero-frequency noise peak gradually develops when the dot-electrode coupling is tuned by gate voltage. As a result, the combination of the current and the shot noise through double quantum dots allows one to probe the presence of Majorana fermions.