{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

6 - i i i i i i 6 The Basic Concept of Bivariate and...

This preview shows pages 1–8. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6 The Basic Concept of Bivariate and Multivariate Random Variables Chapter Outline 6.1 Introduction 6.2 Sample Space 6.3 Joint Cumulative Distribution Function 6.4 Joint Distribution Function 6.5 Marginal Distribution Function 6.6 Conditional Probability 6.6.1 Bivariate Conditional Dis- tribution Function 6.6.2 Multivariate Conditional Distribution Fuction 6.7 Independence 6.7.1 The Independence of Bi- variate Random Variables 6.7.2 The Independence of Mul- tivariate Random Variables 6.8 Linear Correlation 6.8.1 Bivariate: Covariance and Correlation 6.8.2 Multivariate: Covariance Matrix 6.9 Expected Values and Conditional Expected Values 6.9.1 The Expected Value of a Multivariate Function 6.9.2 Conditional Expected Val- ues and Conditional Variance 6.10 Bivariate and Multivariate Proba- bility Models (Supplement) 6.10.1 Bivariate Normal Distribution 6.10.2 Polynominal Distribution 6.10.3 Multivariate Hyper- geometric Distribution 6.10.4 Markov Chains 6.11 The Summary

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6.1 Introduction 6.2 Sample Space S S s s X X Y Y x = X(s) y = Y(s) x y x = X(s) y = Y(s) X, Y (x, y) = (X(s), Y(s)) (a)One dimension (b)Two dimensions Figure 6.1: The diagram of two random variables with the same sample space
6.3 Joint Cumulative Distribution Function Definition 6.1 F X 1 ,...,X n (x 1 , . . . , x n ) = P (X 1 x 1 , X 2 x 2 , . . . , X n x n ), x 1 , x 2 , . . . x n R Bivariate Joint Cumulative Distribution Function Definition 6.2 The joint cumulative distribution Function of two random variables X and Y F X,Y (x, y) = P (X x, Y y), x R , y R The explanation of Definition6.2is as follows: F X,Y (x, y) = P (X x, Y y) is the probability of two events { X x } { Y y } occuring at the same time. It adapts to all random variables (including discrete random variables and continuous random variables). The cumulative distribution function contains 6 properties as follows:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
1. The cumulative distribution function is non-negative, F X,Y (x, y) 0 , x R , y R ; (6.1) 2. The upper bound of the cumulative distribution func- tion is 1, F X,Y (x, y) 1 , x R , y R ; (6.2) 3. The cumulative distribution function is a non-decreasing function, F X,Y (x 1 , y 1 ) F X,Y (x 2 , y 2 ), x 1 x 2 , y 1 y 2 ; (6.3) 4. The cumulative distribution function is a right side continuous function. lim x a + F(x, y) = F(a, y), (6.4) lim y b + F(x, y) = F(x, b) ; (6.5) 5. The limitation of the cumulative distribution function is 1 or 0, lim x →∞ y →∞ F X,Y (x, y) = F( , ) = 1 , (6.6) lim x →−∞ F X,Y (x, y) = F( −∞ , y) = 0 , (6.7) lim y →−∞ F X,Y (x, y) = F(x, −∞ ) = 0 ; (6.8) 6. For any real number a, b, c, andd , when a b and c d , then F(b, d) F(a, d) F(b, c) + F(a, c) 0 (6.9) Theorem 6.1 A function with 6 propertieis (6.1) - (6.9) must be a joint cu- mulative distribution function F X,Y (x, y) of two random vari- ables X and Y .
Example 6.1 Check whether the following function is a joint cumulative distri- bution function.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern