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Unformatted text preview: 1 7. Two Random Variables In many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to record the height and weight of each person in a community or the number of people and the total income in a family, we need two numbers. Let X and Y denote two random variables (r.v) based on a probability model ( , F , P ). Then ( ) = = < 2 1 , ) ( ) ( ) ( ) ( 1 2 2 1 x x X X X dx x f x F x F x X x P and ( ) . ) ( ) ( ) ( ) ( 2 1 1 2 2 1 = = < y y Y Y Y dy y f y F y F y Y y P PILLAI 2 What about the probability that the pair of r.vs ( X , Y ) belongs to an arbitrary region D ? In other words, how does one estimate, for example, Towards this, we define the joint probability distribution function of X and Y to be where x and y are arbitrary real numbers. Properties (i) since we get [ ] ? ) ) ( ( ) ) ( ( 2 1 2 1 = < < y Y y x X x P [ ] , ) , ( ) ) ( ( ) ) ( ( ) , ( = = y Y x X P y Y x X P y x F XY (71) . 1 ) , ( , ) , ( ) , ( = + + = = XY XY XY F x F y F ( ) ( ) , ) ( ) ( , ) ( X y Y X (72) PILLAI 3 Similarly we get (ii) To prove (73), we note that for and the mutually exclusive property of the events on the right side gives which proves (73). Similarly (74) follows. ( ) . ) ( ) , ( = X P y F XY ( ) , ) ( , ) ( = + + Y X . 1 ) ( ) , ( = = P F XY ( ) ). , ( ) , ( ) ( , ) ( 1 2 2 1 y x F y x F y Y x X x P XY XY = < ( ) ). , ( ) , ( ) ( , ) ( 1 2 2 1 y x F y x F y Y y x X P XY XY = < (73) (74) , 1 2 x x > ( ) ( ) ( ) y Y x X x y Y x X y Y x X < = ) ( , ) ( ) ( , ) ( ) ( , ) ( 2 1 1 2 ( ) ( ) ( ) y Y x X x P y Y x X P y Y x X P < + = ) ( , ) ( ) ( , ) ( ) ( , ) ( 2 1 1 2 PILLAI 4 (iii) This is the probability that ( X , Y ) belongs to the rectangle in Fig. 7.1. To prove (75), we can make use of the following identity involving mutually exclusive events on the right side....
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This note was uploaded on 10/16/2009 for the course EL el6303 taught by Professor Prof during the Spring '09 term at NYU Poly.
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