on 26-May-2018 (Sat)

Phoebe was the trusted messenger of Apostle Paul, and she was partly responsible for helping establish a standardized dogma. W

Women like Paula, Marcella, and Fabiola were the driving force behind social services projects that organized religion would eventually become known for -- you know, little things such as monasteries and convents and hospitals for the underprivileg"

In the most notable case, Junia, an early female church figure praised as an apostle,}} was edited into a male in the Middle Ages to reflect a repressive social order.

Before{{c3:: Marcion, }}the Christian Bible as we know it did not exist. There was no known segregation between the{{c3:: "Old" (Jewish) and "New" (Christian) Testaments before}} him. It was just hundreds of different stories in free circulation: multiple books of revelation, too many gospels to count, and more coming in every week. So Marcion decided to go Brothers Grimm on that shit.

Because Marcion believed that the deity of what would eventually be Old Testament texts was a completely different and more{{c4:: malevolent creator-god}}"(

ty)"To make things even more intriguing, one early Gnostic sect called the{{c5:: Carpocratians}} depicted Jesus Christ as not only a sexual being but a{{c6:: full-on libertine who indulged in flamboyantly gay and bisexual antics with his followers}}. That's right, the Gnostic JC was AC/DC"(Chapter:5 Insane Facts That Will Change How You View Christianity)

A new sign of political liberalization appears in China, when the communist government lifts its decade-old ban on the writings of William Shakespeare. The action by the Chinese government was additional evidence that the Cultural Revolution was over

n 1966, Mao Tse-Tung, the leader of the People’s Republic of China, announced a “Cultural Revolution,” which was designed to restore communist revolutionary fervor and vigor to Chinese society. His wife, Chiang Ching, was made the unofficial secretary of culture for China.

a inflação e as tentativas de re- duzir a taxa de juro abaixo da que seria o nível de mercado realizam uma virtual expropriação das dotações dos hospitais, asilos, orfanatos e estabelecimentos similares. Quando os defensores do estado prove- dor lamentam a insuficiência de fundos disponíveis para a assistência humanitária, estão lamentando um dos resultados de políticas que eles mesmos recomendaram

Suponhamos que Paulo, no ano de 1940, tenha poupado cem dó- lares e os tenha aplicado num sistema de previdência social perten- cente ao estado. 11 Em troca, passou a ser credor de algum benefício futuro que lhe deverá ser pago pelo governo. Se o governo gastou os cem dólares em despesas correntes, não houve investimento adi- cional e, portanto, também não houve aumento na produtividade do trabalho. A dívida contraída pelo governo terá de ser paga pe- los futuros contribuintes. Em 1970, um certo Pedro poderá ver-se obrigado a pagar o compromisso assumido pelo governo, embora ele mesmo não tenha auferido nenhum benefício com o fato de Paulo ter poupado cem dólares em 1940

Unlike probability mass functions, probability densities don’t transform quite as naturally along a measurable transformation on X X X . Given a one-to-one transformation g : ℝ N → ℝ N g:RN→RN g : \mathbb{R}^{N} \rightarrow \mathbb{R}^{N} , the differential volumes over which we integrate can change, and probability density functions have to change to compensate. The probability density corresponding to the pushforward distribution along a one-to-one transformation picks up a Jacobian determinant,

π ∗ ( y ) = ∑ n = 1 N π ( x n ( y ) ) ∣ ∣ ∣ ∂ g ∂ x ( x n ( y ) ) ∣ ∣ ∣ − 1 , π∗(y)=∑n=1Nπ(xn(y))|∂g∂x(xn(y))|−1, \pi_{*} (y) = \sum_{n = 1}^{N} \pi(x_{n}(y)) \left| \frac{ \partial g}{\partial x} (x_{n}(y)) \right|^{-1}, where { x 1 ( y ) , … , x N ( y ) } {x1(y),…,xN(y)} \left\{x_{1}(y), \ldots, x_{N}(y) \right\} are the roots of y = g ( x ) y=g(x) y = g(x) for a given y y y .

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Probability Theory (For Scientists and Engineers)

Michael Betancourt

April 2018

Formal probability theory is a rich and complex field of mathematics with a reputation for being confusing if not outright impenetrable. Much of that intimidation, however, is due not to the abstract mathematics but rather how they are employed in practice. In particular, many introductions to probability theory sloppily confound the abstract mathematics with their practical implementations, convoluting what we can calculate in the theory with how we perform those calculations. To make matters even worse, probability theory is used to model a variety of subtlety different systems, which then burdens the already confused mathematics with the distinct and often conflicting philosophical connotations of those applications.

In this case study I attempt to untangle this pedagogical knot to illuminate the basic concepts and manipulations of probability theory. Our ultimate goal is to demystify what we can calculate in probability theory and how we can perform those calculations in practice. We begin with an introduction to abstract set theory, continue to probability theory, and then move onto practical implementations without any interpretational distraction. We will spend time more thoroughly reviewing sampling-based calculation methods before finally considering the classic applications of probability theory and the interpretations of the theory that then arise.

In a few places I will dip a bit deeper into the mathematics than is strictly necessary. These section are labeled “Extra Credit” and may be skipped without any consequence, but they do provide further unification and motivation of some of the more subtle aspects of probability theory.

Let me open with a warning that the section on abstract probability theory will be devoid of any concrete examples. This is not because of any conspiracy to confuse the reader, but rather is a consequence of the fact that we cannot explicitly construct abstract probability distributions in any meaningful sense. Instead we must utilize problem-specific representations of abstract probability distributions which means that concrete examples will have to wait until we introduce these representations in Section 3.

1 Setting A Foundation

Ultimately probability theory concerns itself with sets in a given space. Before we consider the theory at any detail we will first review some of the important results from set theory that we will need. For a more complete and rigorous treatment of set theory see Folland (1999) and the appendix of Lee (2011) .

A set is a collection of elements with a space the collection of all elements under consideration. For example, { 1 } {1} \{1\} , { 5 , 10 , 12 } {5,10,12} \{5, 10, 12\} , and { 30 , 2 , 7 } {30,2,7} \{30, 2, 7\} , are all sets drawn from the space of natural numbers, ℕ = { 0 , 1 , … } N={0,1,…} \mathbb{N} = \left\{0, 1, \ldots \right\} . A point set or atomic set is a set containing a single element such as { 1 } {1} \{1\} above. The entire space itself is always a valid set, as is the empty set or null set, ∅ ∅ \emptyset , which contains no elements at all.

Sets are often defined implicitly via an inclusion criterion. These sets are defined via the notation

A = { x ∈ X ∣ f ( x ) = 0 } , A={x∈X∣f(x)=0}, A = \left\{ x \in X \mid f(x) = 0 \right\}, which reads “the set of elements x x x in the space X X X such that f ( x ) = 0 f(x)=0 f(x) = 0 .” 1.1 Set Operations

There are three natural operations between sets.

Given a set, A A A , its compliment, A c Ac A^{c} , is defined by all of the elements not in that set,

A c = { x ∈ X ∣ x ∉ A } . Ac={x∈X∣x∉A}. A^{c} = \left\{ x \in X \mid x \notin A \right\}.

We can construct the union of two sets, A 1 ∪ A 2 A1∪A2 A_{1} \cup A_{2} , as the set of elements included in ei

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Quem não leva em consideração a escassez de bens de capital disponíveis não é um economista; é um fabulista.

e não fosse a ineficiên- cia do legislador e a lassidão, negligência e corrupção de muitos dos funcionários, os últimos vestígios da economia de mercado já teriam desaparecido há muito tempo