# notes1 - ECE 562 Fall 2011 Gaussian Random Variables and...

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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 σ √ 2 π exp "- ( x- m ) 2 2 σ 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 σ 2 . The statement “ X is Gaussian with mean m and variance σ 2 ” is compactly written as “ X ∼ N ( m,σ 2 ).” The cdf corresponding to the Gaussian pdf is given by F X ( x ) = Z x-∞ f X ( u ) du = Z x-∞ 1 σ √ 2 π exp "- ( u- m ) 2 2 σ 2 # du. This integral cannot be computed in closed-form, but if we make the change of variabe u- m σ = v we get F X ( x ) = Z x- m σ-∞ 1 √ 2 π exp- v 2 2 dv = Φ x- m σ , where Φ is the cdf of a N (0 , 1) random variable, i.e., Φ( x ) = Z x-∞ 1 √ 2 π exp- u 2 2 du....
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notes1 - ECE 562 Fall 2011 Gaussian Random Variables and...

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