Lecture06-2010 - Moment Generating Function and Method of...

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Unformatted text preview: Moment Generating Function and Method of Moments: Lecture VI Charles B. Moss August 31, 2010 I. Moment Generating Function A. Associated with each distribution is a unique function called the moment generating function that can be used to derive the mo- ments of that distribution. 1. Definition 2.17 p-33 The moment generating function M X ( t ) for the random variable X with distribution function f ( x ) is defined as M X ( t ) = E [exp ( tx )] = - exp [ tx ] f ( x ) dt (1) 2. If this moment generating function exists, the moments of the distribution are then defined by r ( x ) = E [ x r ] = M ( r ) X (0) = d r M X ( t ) dt r t =0 (2) 3. As an example, consider the moment generating function for the univariate normal distribution M X ( t ) = 1 2 - exp [ tx ] exp- ( x- ) 2 2 2 dx = 1 2 - exp 2 tx 2 2 2- x 2- 2 x + 2 2 2 dx = 1 2 - exp- x 2 + 2 tx 2 + 2 x- 2 2 2 dx (3) 1 AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture VI Fall 2010 4. Grouping the middle term in the quadratic in we get- x 2 + 2 t 2 + x- 2 (4) 5. To solve this expression, we ask what has to be added, sub- tracted or multiplied to make this expression a perfect square (or quadratic function). t 2 + 2 = t 2 4 + 2 t 2 + 2 (5) 6. Thus, we add and subtract t 2 4 + 2 t 2 M X ( t ) = 1 2 - exp - x 2 + 2 t 2 + x- 2 + t 2 4 + 2 t 2- t 2 4- 2 t 2 2 2 dx = 1 2 - exp - x 2 2 t 2 + x- t 2 + 2 + t 2 4 + 2 t 2 2 2 dx = 1 2 - exp - x- t 2 + 2 2 2 + 1 2 t 2 2 + t dx exp 1 2 t 2 2 + t 1 2 - exp - x- t 2 + 2 2 2...
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Lecture06-2010 - Moment Generating Function and Method of...

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