Analyses of different robustness aspects for models of the direct signal transduction pathway of receptor-induced apoptosis is presented. Apoptosis is a form of programmed cell death, removing unwanted cells within multicellular organisms to maintain a proper balance between cell reproduction and death. Its signalling pathway includes an activation feedback loop that generates bistable behaviour, where the two steady states can be seen as 'life' and 'death'. Inherent robustness, widely recognised in biological systems, is of major importance in apoptosis signalling, as it guarantees the same cell fate for similar conditions. First, the influence of the stochastic nature of reactions indicating a role for inhibition reactions as noise filters and justifying a deterministic approach in the further analyses is evaluated. Second, the robustness of the bistable threshold with respect to parameter changes is evaluated by statistical methods, showing the need to balance both the forward and the back part of the activation loop. These analyses can also discriminate between the models favouring the model consistent with novel biological findings. The parameter robustness analyses are also applicable to other signal transduction networks, as several have been shown to display bistable behaviour. These methods therefore have a range of possible applications in systems biology not only to measure robustness, but also for model discrimination.