Slides06-2010 - Outline Moment Generating Function...

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Unformatted text preview: Outline Moment Generating Function Estimating Probability Functions Moment Generating Function and Method of Moments: Lecture IV Charles B. Moss August 31, 2010 Charles B. Moss Moment Generating Function Outline Moment Generating Function Estimating Probability Functions Moment Generating Function Estimating Probability Functions Subjective Probabilities Objective Estimation of Probability Functions Charles B. Moss Moment Generating Function Outline Moment Generating Function Estimating Probability Functions Moment Generating Function I Associated with each distribution is a unique function called the moment generating function that can be used to derive the moments of that distribution. I 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) I 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) Charles B. Moss Moment Generating Function Outline Moment Generating Function Estimating Probability Functions I 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 tx σ 2 2 σ 2 − x 2 − 2 x μ + μ 2 2 σ 2 dx = 1 σ √ 2 π Z ∞ −∞ exp − x 2 + 2 tx σ 2 + 2 x μ − μ 2 2 σ 2 dx (3) I Grouping the middle term in the quadratic in we get − x 2 + 2 ( t σ 2 + μ ) x − μ 2 (4) Charles B. Moss Moment Generating Function Outline Moment Generating Function Estimating Probability Functions I To solve this expression, we ask what has to be added, subtracted or multiplied to make this expression a perfect square (or quadratic function). ( t σ 2 + μ ) 2 = t 2 σ 4 + 2 t μσ 2 + μ 2 (5) Charles B. Moss Moment Generating Function Outline Moment Generating Function Estimating Probability Functions I Thus, we add and subtract t 2 σ 4 + 2 t μσ 2 M X ( t ) = 1 σ √ 2 π × Z ∞ −∞ exp " − x 2 + 2 ( t σ 2 + μ ) x − μ 2 + t 2 σ 4 + 2 t μσ 2 − t 2 σ 4 − 2 t μσ 2 2 σ 2 # dx = 1 σ √ 2 π Z ∞ −∞ exp " − x 2 2 ( t σ 2 + μ ) x − ( t σ 2 + μ ) 2 + t 2 σ 4 + 2 t μσ 2 2 σ 2 # dx = 1 σ √ 2 π Z ∞ −∞ exp...
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Slides06-2010 - Outline Moment Generating Function...

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