Lecture 06-2007 - LectureVI 1 x2 2 f x = e 2 First,...

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Lecture VI
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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
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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
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Squaring this expression yields ∫ ∫ - - + - - - - - = = dx dy e dx e dy e I x y x y 2 2 2 2 2 2 2 2
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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:
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( 29 ( 29 ( 29 θ θ θ sin cos tan 1 2 2 r x r y y x y x r = = = + = -
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As an example, take Some of the results for this function are: ( 29 2 1 15 2 f x x x = + -
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