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Unformatted text preview: ECE 562 Fall 2011 August 23, 2011 Gaussian Random Variables and Vectors The Gaussian Probability Density Function This is the most important pdf for this course. It also called a normal pdf. f X ( x ) = 1 2 exp " ( x m ) 2 2 2 # . It can be shown this f X integrates to 1 (i.e., it is a valid pdf), and that the mean of the random variable X with the above pdf is m and the variance is 2 . The statement X is Gaussian with mean m and variance 2 is compactly written as X N ( m, 2 ). The cdf corresponding to the Gaussian pdf is given by F X ( x ) = Z x f X ( u ) du = Z x 1 2 exp " ( u m ) 2 2 2 # du. This integral cannot be computed in closedform, but if we make the change of variabe u m = v we get F X ( x ) = Z x m  1 2 exp v 2 2 dv = x m , where is the cdf of a N (0 , 1) random variable, i.e., ( x ) = Z x 1 2 exp u 2 2 du....
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This document was uploaded on 02/08/2012.
 Fall '09

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