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7_mgf - 2 Chapter 7 The Transformation of Random...

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2 Chapter 7 The Transformation of Random Variables (Supplement) 7 The Transformation of Random Variables (Supplement) Chapter Outline 7.1 Introduction 7.2 The Transformation via the Dis- tribution Function 7.2.1 The Transformation of Univariate Discrete Ran- dom Variables 7.2.2 The Transformation of Univariate Continuous Random Variables 7.2.3 The Transformation of Bi- variate Discrete Random Variables 7.2.4 The Transformation of Bi- variate Continuous Ran- dom Variables 7.2.5 The Transformation of the Function of Non-one-to- one Random Variables 7.3 The Moment Generating Func- tion of Random Variables 7.3.1 Function 1 of the Moment Generating Function: Veri- fying Distributions 7.3.2 Function 2 of the Moment Generating Function: Gen- erating Moments 7.4 The Convergent Properties of Random Variables 7.4.1 A Random Variable Con- verges to a Random Variable 7.4.2 A Random Variable Con- verges to a Real Number 7.4.3 The Law of Large Number 7.5 The Key Point
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7.1 Introduction 3 7.1 Introduction 7.2 The Transformation via the Distribution Function 7.2.1 The Transformation of Univariate Discrete Random Variables Theorem 7.1 If X is a discrete random variable with the probability mass function f X ( x ) , and Y = u ( X ) is a one-to-one function, then the probability mass function of Y is f Y ( y ) = f X u - 1 ( y ) squaresolid
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4 Chapter 7 The Transformation of Random Variables (Supplement) Example 7.1 If X geometric ( p = 1 10 ) , then the probability mass function is f X ( x ) = 1 10 ( 9 10 ) x - 1 , x = 1 , 2 , . . . 0 , otherwise Let Y = X 2 , compute the probability mass function f Y ( y ) of Y . 7.2.2 The Transformation of Univariate Continuous Random Vari- ables Theorem 7.2 If X is a continuous random variable with the probability den- sity function f X ( x ) , and Y = u ( X ) is a one-to-one function, then the probability density function of Y f Y ( y ) = f X u - 1 ( y ) | J | , where | J | is the absolute value of J and J = dx / dy is called as Jacobian
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7.2 The Transformation via the Distribution Function 5 7.2.3 The Transformation of Bivariate Discrete Random Variables Theorem 7.3 If X 1 and X 2 are bivariate discrete random variables with the joint probability mass function f X 1 , X 2 ( x 1 , x 2 ) , and Y 1 = u 1 ( X 1 , X 2 ), Y 2 = u 2 ( X 1 , X 2 ) are one-to-one functions, then the joint probability mass function of Y 1 and Y 2 is f Y 1 , Y 2 ( y 1 , y 2 ) = f X 1 , X 2 u - 1 1 ( y 1 , y 2 ), u - 1 2 ( y 1 , y 2 ) Example 7.2 Suppose that X 1 and X 2 are the number of traffic accidents in the first highway and in the second highway respectively, and X 1 and X 2 follow poisson distributions with parameters μ 1 and μ 2 respectively. Let Y 1 is total number of traffic accidents from two highways. Compute P ( Y 1 > 1 ) .
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6 Chapter 7 The Transformation of Random Variables (Supplement) 7.2.4 The Transformation of Bivariate Continuous Random Vari- ables Theorem 7.4 If X 1 and X 2 are two continuous random variables with the joint probability density function f X 1 , X 2 ( x 1 , x 2 ) , and Y 1 = u 1 ( X 1 , X 2 ), Y 2 = u 2 ( X 1 , X 2 ) are one-to-one functions, then the joint probability density function of Y 1 and Y 2 is f Y 1 , Y 2 ( y 1 , y 2 ) = f X 1 , X 2 u - 1 1 ( y 1 , y 2 ), u - 1 2 ( y 1 , y 2 ) | J | ,
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