# 4.09 - 1 Chapter 4 Section 9 Continuous Random Variables...

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Unformatted text preview: 1 Chapter 4, Section 9 Continuous Random Variables Moment-Generating Functions John J Currano, 03/15/2010 2 Definitions. Let Y be a continuous random variable with density function, f ( y ), and let k = 1, 2, 3, … . Then: [ ] ∫ ∞ ∞- = = ′ dy y f y Y E k k k ) ( μ ( 29 [ ] ( 29 [ ] k k k Y E Y E Y E μ μ- =- = ) ( is the k th moment of Y ( about the origin ) . is the k th central moment of Y . Notes. μ = E ( Y ) = the first moment of Y ( = μ 1 ′ ) μ 1 = E ( Y – μ ) = σ 2 = V ( Y ) = the second central moment of Y ( = μ 2 ) = E ( Y 2 ) – [ E ( Y )] 2 = μ 2 ′ – ( μ 1 ′ ) 2 . 3 Definition. If Y is a continuous random variable, its moment- generating function ( mgf ) is the function ( 29 dy y f e e E t m ty tY Y ∫ ∞ ∞- = = ) ( ) ( provided this function of t exists (converges) in some interval around 0 (“there exists a constant b > 0 such that m ( t ) is finite for | t | ≤ b ”)....
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4.09 - 1 Chapter 4 Section 9 Continuous Random Variables...

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