Random Variables and Univariate Probability Distributions Cumulative Distribution Function Cumulative Distribution Function Definition 3(3.3).Cumulative Distribution Function(CDF)] The CDF of a random variable X is defined as: Fx(x)=P(X≤c)for all x∈R. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 31/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 31/287 Definition 3 (3.3). [ Cumulative Distribution Function(CDF) ] Cumulative Distribution Function Cumulative Distribution Function
Random Variables and Univariate Probability Distributions Cumulative Distribution Function Cumulative Distribution Function Theorem 2 (3.2).Properties of Fx() Suppose Fx()is the CDF of some random variable X.Then (1)limx→-oFx(x)=0,limx→+ooFx(c)=1; (2)Fx(x)is non-decreasing,i.e.,for any 1<x2,Fx(x1)< Fx(x2); (3)Fx(x)is right-continuous,i.e.,for all z and 6 >0, giD[Ex(x+o）-Fx(x刃=0. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 32/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 32/287 Theorem 2 (3.2). [ Properties of 𝑭𝑿(∙) ] Cumulative Distribution Function Cumulative Distribution Function
Random Variables and Univariate Probability Distributions Cumulative Distribution Function Cumulative Distribution Function Theorem 3 (3.3) Let a <b.Then P(a<X<6)=Fx(6)-Fx(a). Proof: Observing that the event{X≤b}={X≤a}U{a<X≤b} and the events ix af and fa <x <b are disjoint,we have P(X≤b)=P(X≤a)+P(a<X≤b) Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 33/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 33/287 Theorem 3 (3.3) Cumulative Distribution Function Cumulative Distribution Function Proof:
Random Variables and Univariate Probability Distributions Cumulative Distribution Function Cumulative Distribution Function Theorem 4 (3.4) P(X>6)=1-Fx(6). Proof: The result follows immediately from P(A)=1-P(A)and the definition of CDF Fx(x),with A=x <a. Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 34/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 34/287 Theorem 4 (3.4) Cumulative Distribution Function Cumulative Distribution Function Proof:
Random Variables and Univariate Probability Distributions Cumulative Distribution Function Cumulative Distribution Function Example 7(3.7) Suppose F(x)is a CDF.Define G(x)=1-F(-x).Is G(x)a CDF too? Solution Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May23,2019 35/287
Random Variables and Univariate Probability Distributions Random Variables and Univariate Probability Distributions Introduction to Statistics and Econometrics May 23, 2019 35/287 Example 7 (3.7) Cumulative Distribution Function Cumulative Distribution Function Solution