# Category Introduction to the Mathematical and Statistical Foundations of Econometrics

## Quality Control in Practice

The problem in applying this result in quality control is that K is unknown. Therefore, in practice the following decision rule as to whether K < R or not is followed. Given a particular number r < n, to be determined at the end of this subsection, assume that the set of N bulbs meets the minimum quality requirement K < R if the number k of defective bulbs in the sample is less than or equal to r. Then the set A(r) = {0, 1,…, r} corresponds to the assumption that the set of N bulbs meets the minimum quality requirement K < R, hereafter indicated by “accept,” with probability

r

P(A(r)) = £ P({k}) = pr(n, K), (1.12)

k=0

say, whereas its complement A(r) = {r + 1n} corresponds to the assump­tion that this set of N bulbs does not meet this quality requirement, hereafter indicated by “reject...

## Sampling with Replacement

As a third example, consider the quality control example in the previous section except that now the light bulbs are sampled with replacement: After a bulb is tested, it is put back in the stock of N bulbs even if the bulb involved proves to be defective. The rationale for this behavior may be that the customers will at most accept a fraction R/N of defective bulbs and thus will not complain as long as the actual fraction K/N of defective bulbs does not exceed R /N. In other words, why not sell defective light bulbs if doing so is acceptable to the customers?

The sample space ^ and the a-algebra & are the same as in the case of sampling without replacement, but the probability measure P is different. Con­sider again a sample Sj of size n containing k defective light bulbs...

## Why Do We Need Sigma-Algebras of Events?

In principle we could define a probability measure on an algebra Ж of sub­sets of the sample space rather than on a a-algebra. We only need to change condition (1.10) as follows: For disjoint sets Aj є Ж such that Uj= Aj є Ж, P(UjO0=1 Aj) = Yj=1 P(Aj). By letting all but a finite number of these sets be equal to the empty set, this condition then reads as follows: For disjoint sets Aj є Ж, j = 1, 2,…,n < to, P(Un=1 Aj) = =1 P(Aj). However, if

we confined a probability measure to an algebra, all kinds of useful results would no longer apply. One of these results is the so-called strong law of large numbers (see Chapter 6).

As an example, consider the following game...

## Properties of Algebras and Sigma-Algebras

1.4.1. General Properties

In this section I will review the most important results regarding algebras, a – algebras, and probability measures.

Our first result is trivial:

Theorem 1.1: If an algebra contains only a finite number of sets, then it is a a-algebra. Consequently, an algebra of subsets of a finite set ^ is a a-algebra.

However, an algebra of subsets of an infinite set ^ is not necessarily a a – algebra. A counterexample is the collection &+ of all subsets of ^ = (0, 1] of the type (a, b], where a < b are rational numbers in [0, 1] together with their finite unions and the empty set 0. Verify that &+ is an algebra. Next, let pn = [10nn]/10n and an = 1/pn, where [x] means truncation to the near­est integer < x. Note that pn n; hence, an I n-1 as n ^ ж...