Another way of explaining the same idea invokes the notion of a pivotal quantity.  Because of how the above statistical model is set up, we can construct the pivotal quantity \(\hat{\theta}(s) - \theta\). We know the distribution of this quantity, namely normal and with the aforementioned variance. Moreover, this distribution is independent of the sample, and it is such that fixing the sample to \(s\), and so fixing the value of \(\hat{\theta}\), uniquely determines a distribution over the parameter values \(\theta\). The fiducial argument thus allows us to construct a probability distribution over the parameter values on the basis of the observed sample. The argument can be run whenever we can construct a pivotal quantity like that or, equivalently, whenever we can express the statistical model as a functional model.