The postsynaptic density (PSD) is a cytoskeletal specialization within the postsynaptic membrane of a neuron that helps to concentrate and organize neurotransmitter receptors at a chemical synapse. The total number of receptors within the PSD, which is a major factor in determining the physiological strength or weight of a synapse, fluctuates due to the surface diffusion of receptors into and out of the PSD, and the interactions of receptors with scaffolding proteins and cytoskeletal elements within the PSD. In this article, we present a stochastic model of protein receptor trafficking at the PSD that takes into account these various processes. The PSD is treated as a stochastically gated corral, which contributes a source of extrinsic or environmental noise that supplements the intrinsic noise arising from small receptor numbers. Using a combination of stochastic analysis and Monte Carlo simulations, we determine the time-dependent variation in the mean and variance of synaptic receptor numbers for a variety of initial conditions that simulate fluorescence recovery after photobleaching experiments, and indicate how such data might be used to infer certain properties of the PSD.