Chapter 4
Some Elementary Statistical
Inferences
4.1 SamplingandStatistics
In Chapter 2, we introduced the concepts of samples and statistics. We continue
with this development in this chapter while introducing the main tools of inference:
confidence intervals and tests of hypotheses.
In a typical statistical problem, we have a random variableXof interest, but its
pdff(x)orpmfp(x) is not known. Our ignorance aboutf(x)orp(x) can roughly
be classified in one of two ways:
1.f(x)orp(x) is completely unknown.- The form off(x)orp(x) is known down to a parameterθ,whereθmay be a
vector.
For now, we consider the second classification, although some of our discussion
pertains to the first classification also. Some examples are the following:
(a)Xhas an exponential distribution, Exp(θ), (3.3.6), whereθis unknown.(b)X has a binomial distributionb(n, p), (3.1.2), wherenis known butpis
unknown.(c)Xhas a gamma distribution Γ(α, β), (3.3.2), where bothαandβare unknown.(d)Xhas a normal distributionN(μ, σ^2 ), (3.4.6), where both the meanμand
the varianceσ^2 ofXare unknown.We often denote this problem by saying that the random variableXhas a density
or mass function of the formf(x;θ)orp(x;θ), whereθ∈Ω for a specified set Ω. For
example, in (a) above, Ω ={θ|θ> 0 }.Wecallθa parameter of the distribution.
Becauseθis unknown, we want to estimate it.
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