# lect7 - 7 Two Random Variables In many experiments the...

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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 ξ and . ) ( ) ( ) ( ) ( 2 1 1 2 2 1 = = < y y Y Y Y dy y f y F y F y Y y P ξ PILLAI

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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 ξ ξ , 0 ) , ( ) ) ( ( ) ) ( ( ) , ( = = y Y x X P y Y x X P y x F XY ξ ξ (7-1) . 1 ) , ( , 0 ) , ( ) , ( = +∞ +∞ = −∞ = −∞ XY XY XY F x F y F () ( ) , ) ( ) ( , ) ( −∞ −∞ ≤ξ ξ ξ X y Y X (7-2) PILLAI
3 Similarly we get (ii) To prove (7-3), we note that for and the mutually exclusive property of the events on the right side gives which proves (7-3). Similarly (7-4) follows. () . 0 ) ( ) , ( = −∞ −∞ ξ X P y F XY , ) ( , ) ( = +∞ +∞ ≤ξ ξ Y X . 1 ) ( ) , ( = = P F XY ). , ( ) , ( ) ( , ) ( 1 2 2 1 y x F y x F y Y x X x P XY XY = ξ ). , ( ) , ( ) ( , ) ( 1 2 2 1 y x F y x F y Y y x X P XY XY = < ξ (7-3) (7-4) , 1 2 x x > ( ) y Y x X x y Y x X y Y x X < = ) ( , ) ( ) ( , ) ( ) ( , ) ( 2 1 1 2 ξ ξ ξ ξ ξ ξ y Y x X x P y Y x X P y Y x X P < + = ) ( , ) ( ) ( , ) ( ) ( , ) ( 2 1 1 2 ξ ξ ξ ξ ξ ξ PILLAI

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4 (iii) This is the probability that ( X , Y ) belongs to the rectangle in Fig. 7.1. To prove (7-5), we can make use of the following identity involving mutually exclusive events on the right side. () ). , ( ) , ( ) , ( ) , ( ) ( , ) ( 1 1 2 1 1 2 2 2 2 1 2 1 y x F y x F y x F y x F y Y y x X x P XY XY XY XY + = < ξ (7-5) 0 R ( ) . ) ( , ) ( ) ( , ) ( ) ( , ) ( 2 1 2 1 1 2 1 2 2 1 y Y y x X x y Y x X x y Y x X x < < < = ξ ξ ξ ξ ξ 1 y 2 y 1 x 2 x X Y Fig. 7.1 0 R PILLAI
5 () ( ) 2 1 2 1 1 2 1 2 2 1 ) ( , ) ( ) ( , ) ( ) ( , ) ( y Y y x X x P y Y x X x P y Y x X x P < < + < = ξ ξ ξ ξ ξ 2 y y = 1 y .

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## This note was uploaded on 10/01/2009 for the course ECE 2521 taught by Professor Jacobs during the Fall '09 term at Pittsburgh.

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lect7 - 7 Two Random Variables In many experiments the...

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