2.12 Normal mean and variance both unknown
2.12.1 Formulation of the problem
It is much more realistic to suppose that both parameters of a normal distribution are unknown rather than just one. So we consider the case where we have a set of observations which are with θ and both unknown. Clearly,
from which it follows that the density is in the two-parameter exponential family as defined above. Further
wherein we define
(rather than as as in the case where the mean is known to be equal to μ). It is also convenient to define
(rather than s2=S/n as in the case where the mean is known).
It is worth noting that the two-dimensional vector or equivalently is clearly sufficient for given ...