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

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i i i 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

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i i i 6.1 Introduction 6.2 Sample Space S S s s X X Y Y x = X(s) y = Y(s) x y x = y = 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
i i i 6.3 Joint Cumulative Distribution Function Defnition 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 Defnition 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 De±nition6.2is as follows: F X,Y (x, y) = P (X 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:

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i i i 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 R ; (6.2) 3. The cumulative distribution function is a non-decreasing function, F X,Y (x 1 1 ) F X,Y (x 2 2 ), x 1 x 2 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,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) = −∞ ,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 .
i i i Example 6.1 Check whether the following function is a joint cumulative distri- bution function.

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

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