Lecture 06-2007 - Lecture VI The order of proof of the...

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Unformatted text preview: Lecture VI The order of proof of the normal distribution function is to start with the standard normal: ( 29 2 2 2 1 x e x f- = π ◦ 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. ◦ 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 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. ◦ By polar space, any point ( x , y ) can be written in a trigonometric form: ( 29 ( 29 ( 29 θ θ θ sin cos tan 1 2 2 r x r y y x y x r = = = + =- ◦ As an example, take Some of the results for this function are: ( 29 2 1 15 2 f x x x = +- x...
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This note was uploaded on 11/08/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

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Lecture 06-2007 - Lecture VI The order of proof of the...

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