Lecture 07-functions of RV

Lecture 07-functions of RV - 1036: Probability &...

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Prob. & Stat. Lecture07 - functions of RVs (cwliu@twins.ee.nctu.edu.tw) 7-1 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 7 Lecture 7 Functions of Functions of Random Variables Random Variables
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Prob. & Stat. Lecture07 - functions of RVs (cwliu@twins.ee.nctu.edu.tw) 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 = = = = = =
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Prob. & Stat. Lecture07 - functions of RVs (cwliu@twins.ee.nctu.edu.tw) 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
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Prob. & Stat. Lecture07 - functions of RVs (cwliu@twins.ee.nctu.edu.tw) 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 = = = = = = =
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Prob. & Stat. Lecture07 - functions of RVs (cwliu@twins.ee.nctu.edu.tw) 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
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Lecture 07-functions of RV - 1036: Probability &...

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