Random Variables and Univariate Probability Distributions Random Variables Random Variables However,we can also write C={s∈S:X(s∈A1}U{s∈S:X(s)∈A2} C1UC2,say. The fact that A1 and A2 are disjoint implies that C1 and C2 are also disjoint.It follows that Px(A1UA2)= P(C)+P(C2) Px(A1)+Px(A2). In the rest of this course,we will abuse the notations for the original probability function P()and the induced probability function Px();we will denote both probability functions as P() Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 26/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 26/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables Example 6 (3.6) Suppose we throw three coins.Then the sample space S-HHH,HTH,HHT,THH,THT,TTH,HTT,TTTh. Define a random variable X()to be the number of heads ob- tained from the experiment.Then the new sample space,or the range of X,is given by 2={0,1,2,3}. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 27/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 27/287 Example 6 (3.6) Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables Example 6(3.6) Now,suppose we are interested in calculating the probability that P(0≤X≤1).Denote C={s∈S:0≤X(s)≤1} TTT,TTH,THT,HTTY It follows that P(0≤X≤1)=P(C) PTTT+P(TTH)+PTHT)+(HTT) 1 2 Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 28/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 28/287 Example 6 (3.6) Random Variables Random Variables
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 29/287
Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 29/287 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 Cumulative Distribution Function Cumulative Distribution Function Question: How to characterize a random variable x? Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 30/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 30/287 Cumulative Distribution Function Cumulative Distribution Function Question: How to characterize a random variable 𝑋?