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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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 .
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 15

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online