Biological neural networks outperform current computer technology in terms of power consumption and computing speed while performing associative tasks, such as pattern recognition. The analogue and massive parallel in-memory computing in biology differs strongly from conventional transistor electronics that rely on the von Neumann architecture. Therefore, novel bio-inspired computing architectures have been attracting a lot of attention in the field of neuromorphic computing. Here, memristive devices, which serve as non-volatile resistive memory, are employed to emulate the plastic behaviour of biological synapses. In particular, CMOS integrated resistive random access memory (RRAM) devices are promising candidates to extend conventional CMOS technology to neuromorphic systems. However, dealing with the inherent stochasticity of resistive switching can be challenging for network performance. In this work, the probabilistic switching is exploited to emulate stochastic plasticity with fully CMOS integrated binary RRAM devices. Two different RRAM technologies with different device variabilities are investigated in detail, and their potential applications in stochastic artificial neural networks (StochANNs) capable of solving MNIST pattern recognition tasks is examined. A mixed-signal implementation with hardware synapses and software neurons combined with numerical simulations shows that the proposed concept of stochastic computing is able to process analogue data with binary memory cells.