The stochastic neuron is a key for event-based probabilistic neural networks. We propose a stochastic neuron using a metal-oxide resistive random-access memory (ReRAM). The ReRAM's conducting filament with built-in stochasticity is used to mimic the neuron's membrane capacitor, which temporally integrates input spikes. A capacitor-less neuron circuit is designed, laid out, and simulated. The output spiking train of the neuron obeys the Poisson distribution. Using the 65-nm CMOS technology node, the area of the neuron is , which is one ninth the size of a 1-pF capacitor. The average power consumption of the neuron is 1.289 W. We introduce the neural array-A modified one-transistor-one-ReRAM (1T1R) crossbar that integrates the ReRAM neurons with ReRAM synapses to form a compact and energy efficient in-memory spiking neural network. A spiking deep belief network (DBN) with a noisy rectified linear unit (NReLU) is trained and mapped to the spiking DBN using the proposed ReRAM neurons. Simulation results show that the ReRAM neuron-based DBN is able to recognize the handwritten digits with 94.7% accuracy and is robust against the ReRAM process variation effect.