Random Variables and Univariate Probability Distributions Random Variables Random Variables Question:Can one show that when s is a countable sam- ple space,the induced probability function Px()satisfies the three axioms of the probability function? Answer:Yes.Please try it! Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 16/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 16/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables When S is continuous and so is uncountable (e.g.S= R),we have to ensure that the set C=sS:X(s)E A}belongs to the o-algebra B,which is associated with the original sample space S.Whether or not C e B depends on the mapping X:S->. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 171287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 17/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables Definition 2 (3.2).Measurable Function A function X:S-R is B-measurable (or measurable with respect to the o-field B generated from S)if for every real number a,the set{s∈S:X(s)≤a}∈B. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 18/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 18/287 Definition 2 (3.2). [ Measurable Function ] Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables Remarks: .A B-measurable function is simply called a measurable function if it does not cause any confusion. .A measurable function ensures that P(X E A)is always well-defined for all subsets A in Bo. If X(.)is not measurable,then there exist subsets in the o-field in R for which probabilities are not defined. In this course,the term "random variable"is restricted to being a B-measurable function from s to R. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 19/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 19/287 Random Variables Random Variables Remarks:
Random Variables and Univariate Probability Distributions Random Variables Random Variables Theorem 1 (3.1) noindent Let B be a g-algebra associated with sample space S. Let f()and g()be B-measurable real valued functions,and c be a real number.Then the functions c.f(),f()+g(),f()g() and |f()|are also B-measurable. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 20/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 20/287 Theorem 1 (3.1) Random Variables Random Variables