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 :-oo<s<oo.Define X(s)-1 if s >0 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: