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