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Chapt 5a

# Chapt 5a - 5 Multiple Random Variables 5.1 5.2 5.3 5.4 5.5...

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5 Multiple Random Variables 5.1 Introduction 167 5.2 Joint CDFs of Bivariate Random Variables 167 5.3 Discrete Random Variables 169 5.4 Continuous Random Variables 173 5.5 Determining Probabilities from a Joint CDF 175 5.6 Conditional Distributions 178 5.7 Covariance and Correlation Coefficient 184 5.8 Many Random Variables 187 5.9 Multinomial Distributions 189 5.10 Chapter Summary 190 5.11 Problems 190 5.1 Introduction We have so far been concerned with the properties of a single random variable defined on a given sample space. Sometimes we encounter problems that deal with two or more random variables defined on the same sample space. In this chapter we consider these multivariate systems. We first consider bivariate ran- dom variables and later consider systems with more than two random variables. 5.2 Joint CDFs of Bivariate Random Variables Consider two random variables X and Y defined on the same sample space. For example, X can denote the grade of a student and Y can denote the height of the same student. The joint cumulative distribution function (joint CDF) of X and Y 167

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168 Chapter 5 Multiple Random Variables is given by F XY ( x , y ) = P [ X x , Y y ] The pair ( X , Y ) is referred to as a bivariate random variable. If we define F X ( x ) = P [ X x ] as the marginal CDF of X and F Y ( y ) = P [ Y y ] as the marginal CDF of Y , then we define the random variables X and Y to be independent if F XY ( x , y ) = F X ( x ) F Y ( y ) for every value of x and y . 5.2.1 Properties of the Joint CDF As a probability function F XY ( x , y ) has certain properties, which include the fol- lowing: 1. Since F XY ( x , y ) is a probability, 0 F XY ( x , y ) 1 for −∞ < x < , −∞ < y < . 2. If x 1 x 2 and y 1 y 2 , then F XY ( x 1 , y 1 ) F XY ( x 2 , y 1 ) F XY ( x 2 , y 2 ) . Simi- larly, F XY ( x 1 , y 1 ) F XY ( x 1 , y 2 ) F XY ( x 2 , y 2 ) . This follows from the fact that F XY ( x , y ) is a nondecreasing function of x and y . 3. lim x →∞ y →∞ F XY ( x , y ) = F XY ( , ) = 1 4. lim x →−∞ F XY ( x , y ) = F XY ( −∞ , y ) = 0 5. lim y →−∞ F XY ( x , y ) = F XY ( x , −∞ ) = 0 6. lim x a + F XY ( x , y ) = F XY ( a , y ) 7. lim y b + F XY ( x , y ) = F XY ( x , b ) 8. P [ x 1 < X x 2 , Y y ] = F XY ( x 2 , y ) F XY ( x 1 , y ) 9. P [ X x , y 1 < Y y 2 ] = F XY ( x , y 2 ) F XY ( x , y 1 ) 10. If x 1 x 2 and y 1 y 2 , then P [ x 1 < X x 2 , y 1 < Y y 2 ] = F XY ( x 2 , y 2 ) F XY ( x 1 , y 2 ) F XY ( x 2 , y 1 ) + F XY ( x 1 , y 1 ) 0 Also, the marginal CDFs are obtained as follows: F X ( x ) = F XY ( x , ) F Y ( y ) = F XY ( , y ) From the above properties we can answer questions about X and Y .
5.3 Discrete Random Variables 169 Example 5.1 Given two random variables X and Y with the joint CDF F XY ( x , y ) and marginal CDFs F X ( x ) and F Y ( y ) , respectively, compute the joint probability that X is greater than a and Y is greater than b .

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Chapt 5a - 5 Multiple Random Variables 5.1 5.2 5.3 5.4 5.5...

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