Normal sampling theory: estimation of ‘parameters’ by confidence intervals, by maximum likelihood

Abstract

We have seen in Section 4.2 that a normal population is completely specified if its mean μ and standard deviation σ are known. For this reason μ and σ are often called ‘parameters’ of the normal distribution. In general, if the values of certain properties of a theoretical distribution (e.g. its mean, variance, 3rd moment, etc.) are known and if the distribution is then completely specified (i.e. if its frequency curve can be plotted) these properties are called ‘parameters’ of the distribution. The number of such properties required to be known before a distribution is completely specified is called ‘the number of parameters of the distribution’. Thus the normal distribution has two parameters. But these are not necessarily μ and σ, and, strictly speaking, any two properties that will specify a normal distribution are entitled to be called its parameters (for example, if we know the values of μ + σ and μ - σ, the distribution is specified, because the values of μ and σ would in fact, though not in appearance, be known; again, μ and v₃, see Section 4.4, can act as parameters).