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Unformatted text preview: Chapter 6 Jointly Distributed Random Variables 6.1 Joint Distribution Functions Definition 6.1 For any two random variables X and Y , the joint cumulative distribution function ( joint c.d.f. ) of X and Y is defined by F ( a,b ) = P ( X ≤ a,Y ≤ b ) ∞ < a,b < ∞ • F X ( a ) = P ( X ≤ a ) = lim b →∞ F ( a,b ) = F ( a, ∞ ) • F Y ( b ) = P ( Y ≤ b ) = lim a →∞ F ( a,b ) = F ( ∞ ,a ) • F X and F Y are referred to as the marginal distributions of X and Y . Definition 6.2 The joint probability density function of discrete random variables X and Y is defined by f ( x,y ) = P ( X = x,Y = y ) • f X ( x ) = P ( X = x ) = X y f ( x,y ) • f Y ( y ) = P ( Y = y ) = X x f ( x,y ) • f X and f Y are called the marginal density functions of X and Y . Example 6.1 3 balls are randomly selected, without replacement, from an urn containing 3 red, 4 white, and 5 blue balls. Let X and Y denote, respectively, the number of red and white balls chosen. Find the joint probability density function of X and Y . 6-1 Solution : x \ y 1 2 3 P ( X = x ) 10 220 40 220 30 220 4 220 1 30 220 60 220 18 220 2 15 220 12 220 3 1 220 P ( Y = y ) Properties: (i) f ( x,y ) ≥ (ii) X x X y f ( x,y ) = 1 (iii) For any region A in the xy-plane, P (( X,Y ) ∈ A ) = ∑∑ ( x,y ) ∈ A f ( x,y ) Example 6.2 The joint probability density function of X and Y is given by f ( x,y ) = cxy, for x = 1 , 2 , 3; y = 1 , 2 , 3 ....
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