Lecture-17

Lecture-17 - Lecture 17. The Normal Distribution Fact:...

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Unformatted text preview: Lecture 17. The Normal Distribution Fact: integraldisplay - e- x 2 / 2 dx = 2 Proof: parenleftBigg integraldisplay - e- x 2 / 2 dx parenrightBigg 2 = parenleftbiggintegraldisplay - e- x 2 / 2 dx parenrightbiggparenleftbiggintegraldisplay - e- y 2 / 2 dy parenrightbigg = integraldisplay - parenleftbiggintegraldisplay - e- x 2 / 2 dx parenrightbigg e- y 2 / 2 dy = integraldisplay - integraldisplay - e- x 2 / 2 e- y 2 / 2 dx dy = integraldisplay - integraldisplay - e- ( x 2 + y 2 ) / 2 dx dy = integraldisplay 2 integraldisplay e- r 2 / 2 r dr d = 2 integraldisplay e- r 2 / 2 r dr = 2 integraldisplay-- e u du (letting u =- r 2 / 2) = 2 ///// From the fact above, we can conclude that 1 2 e- x 2 / 2 is a pdf, since this is a non-negative, continuous function whose integral over the line is 1. A random variable having this as its pdf is called standard normal. Often the letter Z is used to denote such a variable.is used to denote such a variable....
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This note was uploaded on 11/29/2011 for the course MATH 3355 taught by Professor Britt during the Spring '08 term at LSU.

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Lecture-17 - Lecture 17. The Normal Distribution Fact:...

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