lecture7 - Lecture-7 Two Random Variables In many...

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1 Lecture-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 and (29 = - = < 2 1 , ) ( ) ( ) ( ) ( 1 2 2 1 x x X X X dx x f x F x F x X x P ξ . ) ( ) ( ) ( ) ( 2 1 1 2 2 1 = - = < y y Y Y Y dy y f y F y F y Y y P
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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 , 0 ) , ( ) ) ( ( ) ) ( ( ) , ( = = y Y x X P y Y x X P y x F XY (1) . 1 ) , ( , 0 ) , ( ) , ( = +∞ +∞ = -∞ = -∞ XY XY XY F x F y F (29 ( 29 , ) ( ) ( , ) ( -∞ -∞ X y Y X (2)
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3 Similarly we get (ii) To prove (3), we note that for and the mutually exclusive property of the events on the right side gives which proves (3). Similarly (4) follows. (29 . 0 ) ( ) , ( = -∞ -∞ ξ 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 - = < (3) (4) , 1 2 x x ( 29 ( 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
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4 (iii) This is the probability that ( X , Y ) belongs to the rectangle in Fig. 7.1. To prove (5), we can make use of the following identity involving mutually exclusive events on the right side. (29 ). , ( ) , ( ) , ( ) , ( ) ( , ) ( 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 (5) ( 29 ( . ) ( , ) ( ) ( , ) ( ) ( , ) ( 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 < < < = < 0 R 1 y 2 y 1 x 2 x X Y Fig. 7.1 0 R
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5 (29 ( 29 ( 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 2 y y = 1 y This gives and the desired result in (5) follows by making use of (3) with and respectively.
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lecture7 - Lecture-7 Two Random Variables In many...

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