Hardware implementations of artificial neural networks typically require significant amounts of hardware resources. This paper proposes a novel radial basis function artificial neural network using stochastic computing elements, which greatly reduces the required hardware. The Gaussian function used for the radial basis function is implemented with a two-dimensional finite state machine. The norm between the input data and the center point is optimized using simple logic gates. Results from two pattern recognition case studies, the standard Iris flower and the MICR font benchmarks, show that the difference of the average mean squared error between the proposed stochastic network and the corresponding traditional deterministic network is only 1.3% when the stochastic stream length is 10kbits. The accuracy of the recognition rate varies depending on the stream length, which gives the designer tremendous flexibility to tradeoff speed, power, and accuracy. From the FPGA implementation results, the hardware resource requirement of the proposed stochastic hidden neuron is only a few percent of the hardware requirement of the corresponding deterministic hidden neuron. The proposed stochastic network can be expanded to larger scale networks for complex tasks with simple hardware architectures.