radically elementary probability theory - edward nelson

Ngày tải lên: 31/03/2014, 16:25

... 1 Introduction to Probability Theory 1.1 The Binomial Asset Pricing Model The binomial asset pricing model provides a powerful tool to understand arbitrage pricing theory and probability theory. In ... half-lines a; 1= 1  n=1 a; a + n and ,1;a= 1  n=1 a , n; a are Borel. Half-open and half-closed intervals are also Borel, since they can be written as intersec- tions of open...

Ngày tải lên: 18/10/2013, 03:20

Ngày tải lên: 17/12/2013, 15:18

... beautiful and provide tools that yield powerful algorithms for solving problems in number theory. For ex- ample, continued fractions provide a fast way to write a prime—even a hundred digit prime—as ... instructions that can be fol- lowed to perform a specific task, such as a sequence of instructions in a computer program, which must terminate on any valid input. The word “al- gorithm” is som...

Ngày tải lên: 07/03/2014, 16:20

Ngày tải lên: 31/03/2014, 16:21

... < b, b < a. 6. The Well-Ordering Property for N: Every non-empty subset of N contains a least element. 7. The Principle of Mathematical Induction: Let P (n)beastate- ment concerning the integer ... are per- mitted, provided that all copies and derivatives retain the same permissions. Specifically no commerical use of these notes or any revisions thereof is per- mitted. i 12 CHAPTER 3. E...

Ngày tải lên: 31/03/2014, 16:21

Ngày tải lên: 31/03/2014, 16:21

... pseudo- prime to base a (or an a-pseudoprime) if a n−1 ≡ 1 mod n. Fermat’s Little Theorem can be restated as Proposition 5.1 If n is an odd prime then it is a pseudoprime to all bases a co- prime ... numbers N are well-ordered, ie every subset S ⊂ N has a least element. 1.4 The Fundamental Theorem of Arithmetic Proposition 1.4 (Euclid’s Lemma) Suppose p ∈ N is a prime number; and sup- pose a,...

Ngày tải lên: 31/03/2014, 16:21

... expanding, x 1 + x 2 2 ≥ √ x 1 x 2 , (1.2) which is the Arithmetic-Mean-Geometric-Mean Inequality for n = 2. Assume that the Arithmetic-Mean-Geometric-Mean Inequality holds true for n = 2 k−1 , k > 2, ... (y 1 y 2 ···y 2 k ) 1/2 k . (1.4) ❑ This means that the 2 k−1 -th step implies the 2 k -th step, and so we have proved the Arithmetic-Mean-Geometric-Mean Inequality for powers of 2. 10 Ch...

Ngày tải lên: 31/03/2014, 16:21

... 1.2.2. Interpretation: Coin-tossing with one coin, such that one has head with probability p and tail with probability 1 − p. Then µ n,p ({k}) is equals the probability, that within n trials one has k-times head. 1.3. ... :=  C∈J C yields to a σ-algebra according to Proposition 1.1.4 such that (by construc- tion) G ⊆ σ(G). It remains to show that σ(G) is the smallest σ-algebra containin...

Ngày tải lên: 31/03/2014, 16:23