Random Variables and Univariate Probability Distributions Random Variables Random Variables Remarks: If functions X(s)and Y(s)that are measurable map- pings from S to n,then ordinary algebraic operations, such as -Z(s)=ax(s), -Z(s)=X(s)+Y(s), 一 Z(s)=X(s)Y(s),and -Z(s)=X(s)/Y(s): are also measurable. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 21/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 21/287 Random Variables Random Variables Remarks:
Random Variables and Univariate Probability Distributions Random Variables Random Variables If {X1(s),X2(s),..}is a sequence of measurable func- tions,then limiting operations,such as -Z(s)=limi→Z(s）and -Z(s)=limn→sup1<i≤nZ(s川， are also measurable. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 22/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 22/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables The induced probability Px()satisfies the definition of a probability function: Condition (1)of a probability function in Defini- tion 2.11 can be easily verified by observing that for CA={s∈S:X(s)∈A, 1≥Px(A)=P(CA)≥0 given0≤P(CA)≤1 for any CA∈B,where B is a o-field generated from S. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 23/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 23/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables Condition (2)of a probability function also holds for Px(A)because S=is E S:X(s)=}implies Px(2)=P(S)=1. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 24/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 24/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables The induced probability Px()satisfies the definition of a prob- ability function: -For Condition (3)of a probability function,consider two mutually exclusive events A1 and A2 in Bo,a o-field gen- erated from the subsets of Here,the induced probability of A1 UA2 is given by Px(A1UA2)=P(C), where C={s∈S:X(s)∈A1UA2}. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 25/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 25/287 Random Variables Random Variables