Category Introduction to the Mathematical and Statistical Foundations of Econometrics

Probability Measure

Let us return to the Texas lotto example. The odds, or probability, of winning are 1 /N for each valid combination rnj of six numbers; hence, if you play n different valid number combinations {«jl, …,Mjn}, the probability of winning is n/N:P({«^,}) = n/N. Thus, in the Texas lotto case the probability P(A), A e &, is given by the number n of elements in the set A divided by the total number N of elements in ^. In particular we have P(Q) = 1, and if you do not play at all the probability of winning is zero: P(0) = 0.

The function P(A), A e &, is called a probability measure. It assigns a number P(A) e [0, 1] to each set A e &. Not every function that assigns numbers in [0,1] to the sets in & is a probability measure except as set forth in the following definition:

Definition 1...

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Conditional Probability, Bayes’ Rule, and Independence

1.10.1. Conditional Probability

Consider a statistical experiment with probability space { ^, &, P}, and suppose it is known that the outcome of this experiment is contained in a set B with P (B) > 0. What is the probability of an event A given that the outcome of the experiment is contained in B? For example, roll a dice. Then ^ = {1, 2, 3, 4, 5, 6}, & is the a-algebra of all subsets of ^, and P({«}) = 1/6 for rn = 1,2, 3, 4, 5, 6. Let B be the event The outcome is even (B = {2, 4, 6}), and let A = {1, 2, 3 }. If we know that the outcome is even, then we know that the outcomes {1, 3} in A will not occur; if the outcome is contained in A, it is contained in A П B = {2}...

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Quality Control

1.2.1. Sampling without Replacement

As a second example, consider the following case. Suppose you are in charge of quality control in a light bulb factory. Each day N light bulbs are produced. But before they are shipped out to the retailers, the bulbs need to meet a minimum quality standard such as not allowing more than R out of N bulbs to be defective. The only way to verify this exactly is to try all the N bulbs out, but that will be too costly. Therefore, the way quality control is conducted in practice is to randomly draw n bulbs without replacement and to check how many bulbs in this sample are defective.

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As in the Texas lotto case, the number M of different samples Sj of size n you can draw out of a set of N elements without replacement is

Each sample Sj is characterized by ...

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Bayes’ Rule

Let A and B be sets in &. Because the sets A and A form a partition of the sample space ^, we have B = (B П A) U (B П A); hence,

P(B) = P(B П A) + P(B П A) = P(B|A)P(A) + P(B |A)P(A).

Moreover,

P(AB)= P(A П B) P(B|A)P(A)

( 1 ) P (B) P (B) ■

Combining these two results now yields Bayes’ rule:

Подпись: P (A| B)P (B | A) P (A)

P(B | A)P(A) + P(B| A)P(A) ■

Thus, Bayes’ rule enables us to compute the conditional probability P(A |B) if P(A) and the conditional probabilities P(B | A) and P(B | A) are given.

More generally, if Aj, j = 1, 2,…,n (< to) is a partition of the sample space ^ (i. e., the Aj’s are disjoint sets in Ж such that ^ = Un=j Aj), then

Подпись: P(Ai |B)P (B | Ai) P (Ai)

TTj=1 P (B| Aj) P (Aj) •

Bayes’ rule plays an important role in a special branch of statistics (and econometrics) called Bayesian ...

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