Introduction, Set theory, Measure theory, Probability, Random variable

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1. Introduction

probability, random variable, random process, kernel function에 대해서 알아보자

 

2. Set

set, element, subset, universal set, set operations

disjoint

partition

Cartesian product

power set

cardinality |A|: finite, infinite, countable, uncountable, denumerable (countably infinite)

자연수, 실수는 countable, [0,1]사이의 실수집합은 uncountable

mapping, domain, co-domain, image, range, inverse image

one-to-one=injective, onto=surjective, invertible

 

3. Measure theory

σ-algebra = σ-field 정의

properties 특징

Sigma field가 없으면 measure를 정의할 수 없다.

measure 정의

measurable space : $(U,B)$

measure space : $(U, B, \mu)$

 

4. Probability

Probability measure : measure + $\mu(X) = 1$

Bayes' rule

Posterior, Prior probability

independent != disjoint, mutually exclusive

 

5. Random Variable

Random Variable 정의

Probability density function : p.d.f

Conditional expectation, probability $X|Y$

Expectation의 정의

두 분포가 같다는 것은 mean값만 가지고 판단 할 수 없다. n-th momentum을 보고나서 판단한다.(2nd momentum: variance)

Independent => uncorrelated

uncorrelated =\=> independent