Lecture 07-functions of RV

# Lecture 07-functions of RV - 1036 Probability Statistics...

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

Prob. & Stat. Lecture07 - functions of RVs ([email protected]) 7-1 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 7 Lecture 7 Functions of Functions of Random Variables Random Variables

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

View Full Document
Prob. & Stat. Lecture07 - functions of RVs ([email protected]) 7-2 1-1 Transformations of Variable • Suppose that X is a discrete RV with probability function f ( x ). • Suppose further that Y = u ( X ) defines a 1-1 transformation between variables X and Y . • For a 1-1 transformation, the value y = u ( x ) is related to one, and only one, value x=w ( y ), where the function w can be considered the inverse function of u and the value w ( y ) is obtained by solving y = u ( x ), the unique solution, for x terms of y • The probability distribution of Y = u ( X ) is )] ( [ )] ( [ ) ( ) ( y w f y w x X P y Y P y g = = = = = =
Prob. & Stat. Lecture07 - functions of RVs ([email protected]) 7-3 Example 7.1 Xa discrete geometric RV with probability function Find the probability distribution of the RV Y = X 2 . Solution: – Since the values of X are all positive, the transformation defines a one-to-one correspondence between X and Y –H e n c e () ,... 2 , 1 , 0 , 4 1 4 3 ) ( 1 = = x x f x () = = = otherwise , 0 ,... 9 , 4 , 1 , 4 1 4 3 ) ( ) ( 1 y y

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

View Full Document
Prob. & Stat. Lecture07 - functions of RVs ([email protected]) 7-4 1-1 Transformations of Variables • Suppose that X 1 and X 2 are two discrete RVs with probability function f ( x 1 , x 2 ). • Suppose further that Y 1 = u 1 ( X 1 , X 2 ) and Y 2 = u 2 ( X 1 , X 2 ) define a 1-1 transformation between the set of points ( x 1 , x 2 ) and ( y 1 , y 2 ). . • For a 1-1 transformation, the point ( y 1 , y 2 ) is related to one, and only one, point ( x 1 , x 2 ). And the values x 1 = w 1 ( y 1 , y 2 ) and x 2 = w 2 ( y 1 , y 2 ) are the unique solution for the linear equation y 1 = u 1 ( x 1 , x 2 ) and y 2 = u 2 ( x 1 , x 2 ) •T h e f u n c t i o n w 1 and w 2 can be considered the inverse function of u 1 and u 2 , respectively • The joint probability distribution of Y 1 and Y 2 is )] , ( ), , ( [ )] , ( ), , ( [ ) , ( ) , ( 2 1 2 2 1 1 2 1 2 2 2 1 1 1 2 2 1 1 2 1 y y w y y w f y y w X y y w X P y Y y Y P y y g = = = = = = =
Prob. & Stat. Lecture07 - functions of RVs ([email protected]) 7-5 Example 7.2 X 1 and X 2 are two independent discrete RVs with Poisson distributions with parameters µ 1 and µ 2 , respectively. Find the distribution of the random variable Y = X 1 + X 2 . Solution: –S n c e X 1 and X 2 are independent, then – Define a second random variable Y 2 = X 2. . Then the inverse functions are given by x 1 = y 1 – y 2 and x

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 08/23/2009 for the course IEE 1036 taught by Professor Cwliu during the Spring '06 term at National Chiao Tung University.

### Page1 / 22

Lecture 07-functions of RV - 1036 Probability Statistics...

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

View Full Document
Ask a homework question - tutors are online