We analyse a simple gene regulatory system where a protein enhances its own rate of mRNA transcription. For the integer-valued, chemical master equation formulation of the first order reaction network, the first and second moments can be studied analytically. In particular, this allows us to characterise the noise strength in terms of the rate constants. Motivated by the need for efficient multi-scale simulation tools, we then consider a hybrid model where the protein is assumed to be abundant, so that its level may be described by a continuous-valued random variable. This leads to a stochastic differential equation driven by a state-dependent switch, and our aim is to study the extent to which this model can accurately approximate the underlying fully discrete system. We discuss some of the technical difficulties that can arise when such a hybrid system is analysed and simulated and show that the noise strength can be analysed by introducing a discrete-time Euler-Maruyama style approximation and using recent results concerning the convergence of numerical methods for hybrid systems. In this manner, numerical analysis techniques are employed to examine properties of the continuous-time model by treating it as the limit of a discrete-time approximation.