厦门大学：《概率论与数理统计 Probability and Statistics for Economists》课程教学资源（课件讲稿）Chapter 03 Random Variables and Univariate Probability Distributions

誉院究等涛想南型王季大门厦鼻 Random Variables and Univariate Probability Distributions Professor Yongmiao Hong Cornell University May23,2019

Random Variables and Univariate Probability Distributions Professor Yongmiao Hong Cornell University May 23, 2019

CONTENTS 3.1 Random Variables 3.2 Cumulative Distribution Function 3.3 Discrete Random Variables(DRV) 3.4 Continuous Random Variables 3.5 Functions of a Random Variable 3.6 Mathematical Expectation 3.7 Moments 3.8 Quantiles 3.9 Moment Generating Function(MGF) 3.10 Characteristic Function 3.11 Conclusion Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 2/287

Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 2/287 3.1 Random Variables 3.2 Cumulative Distribution Function 3.3 Discrete Random Variables(DRV) 3.4 Continuous Random Variables 3.5 Functions of a Random Variable 3.6 Mathematical Expectation 3.7 Moments 3.8 Quantiles 3.9 Moment Generating Function (MGF) 3.10 Characteristic Function 3.11 Conclusion CONTENTS

Random Variables and Univariate Probability Distributions Random Variables Random Variables It is inconvenient to work with different sample spaces. To develop a unified probability theory,we consider a mapping X from the original sample space S to a new sample space which consists of a set of real numbers. This transformation X:S->is called a random vari- able. In many applications,we may be interested only in some particular aspect of the outcomes of an experiment,rather than the outcomes themselves.A suitably defined ran- dom variable X will better serve for our purpose. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 3/287

Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 3/287 Random Variables Random Variables

Random Variables and Univariate Probability Distributions Random Variables Random Variables Definition 1(3.1).[Random Variable] A random variable,X(),is a B-measurable mapping (or point function)from the sample space S to the real line R such that for each outcome s e S,there exists a corresponding unique real number,X(s).The collection of all possible values that the random variable X can take,also called the range of X(), constitutes a new sample space,denoted as Remark: X:S-need not be a 1-1 mapping.It is possible that X(s1)=X(s2) Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 4/287

Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 4/287 Definition 1 (3.1). [Random Variable] Random Variables Random Variables Remark:

Random Variables and Univariate Probability Distributions Random Variables Random Variables Example 1(3.1) When we throw a coin,the sample space S={H,T.Define a random variable X()by X(H)=1 and X(T)=0.Then we obtain a new sample space ={1,0}. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 5/287

Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 5/287 Example 1 (3.1) Random Variables Random Variables

Random Variables and Univariate Probability Distributions Random Variables Random Variables Example 2 (3.2) For the election of a candidate,the sample space S {Win, Fail}.Define a random variable X()by X(Win)=1 and X(Fail)=0.Then ={1,0. Remark: It is not necessary to have the same number of basic outcomes for both S and Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 6/287

Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 6/287 Example 2 (3.2) Random Variables Random Variables Remark:

Random Variables and Univariate Probability Distributions Random Variables Random Variables Example 3 (3.3) Suppose we throw three fair coins.Then the space sample S-TTT,TTH,THT,HTT,HHT,HTH,THH,HHH. Let X()be the number of heads shown up.Then X(TTT)= 0,X(TTH)=1,X(THT)-1,X(HTT)-1,X(HHT) 2,X(HTH=2,X(THH)=2,X(HHH)=3.We have 2={0,1,2,3}. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 7/287

Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 7/287 Example 3 (3.3) Random Variables Random Variables

Random Variables and Univariate Probability Distributions Random Variables Random Variables Remark: In this example,X(s)denotes the number of heads,and so P(X =3)-P(A),where A={sES:X(s)=3=HHH is the probability that exactly three heads occur. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 8/287

Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 8/287 Random Variables Random Variables Remark:

Random Variables and Univariate Probability Distributions Random Variables Random Variables Example 4 (3.4) When a die is rolled,the sample space S ={1,2,3,4,5,6}. Define X(s)=s.Then =S.This is an identity transforma- tion. Question:Suppose the number of basic outcomes in s is countable.Is it possible that the number of basic outcomes in larger than that of S? Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 9/287

Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 9/287 Example 4 (3.4) Random Variables Random Variables

Random Variables and Univariate Probability Distributions Random Variables Random Variables Example 5(3.5) Suppose S={s :-oo0 andX(s)=0ifs≤0. Remark: Here,X is called a binary random variable because there are only two possible values X can take.The binary variable has wide applications in economics. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 10/287

Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 10/287 Example 5 (3.5) Random Variables Random Variables Remark: