Random Variables and Univariate Probability Distributions Random Variables Random Variables There are many other examples of random variables,including (e.g) subjective well-being, sentiment index of investors, economic policy uncertainty (EPU)index These indices are usually constructed based on text data from social media platforms (e.g.,WeChat,Facebook)and news media.Text data are an unstructured form of Big data. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 11/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 11/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables Definition 3.1 of a random variable is limited to real- valued functions.We could define complex-valued ran- dom variables by looking upon real and imaginary parts separately as two real-valued random variables. Convention:A capital letter X denotes a random vari- able,and a lowercase letter x denotes its realization,i.e., a possible value that random variable X can take (i.e., =X(s)for a given seS). The probability function defined on the original sample space s can be used to obtain the probability distribu- tion of the random variable x. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 12/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 12/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables First,suppose sample space s has a finite number of basic outcomes S={s1,,sn} with a probability function P:B->0,1,where B is a o-field associated with S.Also,a random variable X S→R yields D={c1,,cm} where m may not be the same as n. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 13/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 13/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables ● Then the probability function Px:->R for X can be obtained as Px(xi)= P(X=xi) P(C), where Ci is an event in S such that C={s∈S:X(s)=x} Px()is an induced probability function on the new sam- ple space n,obtained from the original probability func- tion P(). Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 14/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 14/287 Random Variables Random Variables
Random Variables and Univariate Probability Distributions Random Variables Random Variables More formally,for any set AE Bo,where Bo is a o-field generated from we can define a probability function Px:B2→R such that Px(A)-P(CA) Ps∈S:X(s)∈A, where CA={s∈S:X(s)∈A} Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 15/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 15/287 Random Variables Random Variables