Introduction to the Mathematical and Statistical Foundations of Econometrics

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}. Knowing that the outcome is 2, 4, or 6, the probability that the outcome is contained in A is therefore 1/3 = P(A П B)/P(B). This is the conditional probability of A, given B, denoted by P(A|B). If it is revealed that the outcome of a statistical experiment is contained in a particular set B, then the sample space ^ is reduced to B because we then know that the outcomes in the complement of B will not occur, the a-algebra & is reduced to & П B = {A П B, A є &}, the collection of all intersections of the sets in & with B (Exercise: Is this a a-algebra?), and the probability measure involved becomes P(A|B) = P(A П B)/P(B); hence, the probability space becomes {B, &П B, P(-|B)}. See Exercise 19 for this chapter.

Добавить комментарий

Introduction to the Mathematical and Statistical Foundations of Econometrics

Mathematical Expectation

With these new integrals introduced, we can now answer the second question stated at the end of the introduction: How do we define the mathematical ex­pectation if the distribution of …

Hypotheses Testing

Theorem 5.19 is the basis for hypotheses testing in linear regression analysis. First, consider the problem of whether a particular component of the vector Xj of explanatory variables in model …

The Inverse and Transpose of a Matrix

I will now address the question of whether, for a given m x n matrix A, there exists an n x m matrix B such that, with y = Ax, …

Как с нами связаться:

Украина:
г.Александрия
тел./факс +38 05235  77193 Бухгалтерия

+38 050 457 13 30 — Рашид - продажи новинок
e-mail: msd@msd.com.ua
Схема проезда к производственному офису:
Схема проезда к МСД

Партнеры МСД

Контакты для заказов оборудования:

Внимание! На этом сайте большинство материалов - техническая литература в помощь предпринимателю. Так же большинство производственного оборудования сегодня не актуально. Уточнить можно по почте: Эл. почта: msd@msd.com.ua

+38 050 512 1194 Александр
- телефон для консультаций и заказов спец.оборудования, дробилок, уловителей, дражираторов, гереторных насосов и инженерных решений.