Lecture 06-2007 - Derivation of the Normal Distribution:...

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Derivation of the Normal Distribution: Lecture VI I. Derivation of the Normal Distribution Function A. The order of proof of the normal distribution function is to start with the standard normal:   2 2 2 1 x e x f 1. First, we need to demonstrate that the distribution function does integrate to one over the entire sample space, which is  to . This is typically accomplished by proving the constant. 2. Let us start by assuming that dy e I y 2 2 Squaring this expression yields   dx dy e dx e dy e I x y x y 2 2 2 2 2 2 2 2 The trick to this integration is changing the variables into a polar form. 3. Polar Integration: The notion of polar integration is basically one of a change in variables. Specifically, some integrals may be ill-posed in the traditional Cartesian plane, but easily solved in a polar space. a. By polar space, any point   , xy can be written in a trigonometric form:       sin cos tan 1 2 2 r x r y y x y x r i. As an example, take   2 1 15 2 f x x x   . Some of the results for this function are:
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AEB 6933 Mathematical Statistics for Food and Resource Economics
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Lecture 06-2007 - Derivation of the Normal Distribution:...

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