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Unformatted text preview: i i i i i i 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 Nononeto 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 i i 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 onetoone function, then the probability mass function of Y is f Y ( y ) = f X u 1 ( y ) squaresolid i i 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 , . . . , 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 onetoone 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 。 i i 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 onetoone 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 ) . i i 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...
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This note was uploaded on 05/11/2011 for the course IEEM 9834211 taught by Professor Wang during the Spring '09 term at Tsinghua University.
 Spring '09
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