Lecture06-2010

# 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 )] = Z ∞ −∞ exp [ tx ] f ( x ) dt (1) 2. If this moment generating function exists, the moments of the distribution are then defined by μ 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 σ √ 2 π Z ∞ −∞ exp [ tx ] exp " − ( x − μ ) 2 2 σ 2 # dx = 1 σ √ 2 π Z ∞ −∞ exp " 2 txσ 2 2 σ 2 − x 2 − 2 xμ + μ 2 2 σ 2 # dx = 1 σ √ 2 π Z ∞ −∞ exp " − x 2 + 2 txσ 2 + 2 xμ − μ 2 2 σ 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 tσ 2 + μ x − μ 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). tσ 2 + μ 2 = t 2 σ 4 + 2 tμσ 2 + μ 2 (5) 6. Thus, we add and subtract t 2 σ 4 + 2 tμσ 2 M X ( t ) = 1 σ √ 2 π × Z ∞ −∞ exp ⎡ ⎣ − x 2 + 2 tσ 2 + μ x − μ 2 + t 2 σ 4 + 2 tμσ 2 − t 2 σ 4 − 2 tμσ 2 2 σ 2 ⎤ ⎦ dx = 1 σ √ 2 π Z ∞ −∞ exp ⎡ ⎢ ⎣ − x 2 2 tσ 2 + μ x − tσ 2 + μ 2 + t 2 σ 4 + 2 tμσ 2 2 σ 2 ⎤ ⎥ ⎦ dx = 1 σ √ 2 π Z ∞ −∞ exp ⎡ ⎢ ⎣ − x − tσ 2 + μ 2 2 σ 2 + 1 2 t 2 σ 2 + tμ ⎤ ⎥ ⎦ dx exp 1 2 t 2 σ 2 + tμ 1 σ √ 2 π Z ∞...
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Lecture06-2010 - Moment Generating Function and Method of...

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