Consider the manufacture of some component in which a particular measurement, , is of interest. This may be modeled as being a value of a random variable having a Normal distribution with mean and variance , . In this context the mean represents the `setting' of the manufacturing process, while the variance represents the `process variability'.
The parameter is unknown, but prior knowledge about it may be represented by a density, and for this example values of are suitable. The choice of is in fact largely uninformative as a wide range ( say) of values of are all quite likely.
Assume the parameter is known: .
The sample data to hand
can be summarised as .
For this example both analytic and numerical solutions are readily available.