Gaussian+pdf+and+special+properties

Gaussian+pdf+and+special+properties - Gaussian...

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Unformatted text preview: Gaussian distribution: The analytical expression for a simple Gaussian pdf (commonly termed as the standard normal with zero mean and unit variance, i.e. (0,1) X N ∼ ): 2 2 2 2 1 pdf: ( ) , 2 1 cdf: ( ) ( ) ( ) 2 x X x x u X X f x e x F x f u du e du x π π − − −∞ −∞ = − ∞ < < + ∞ = = = Φ ∫ ∫ Note: • the special function ( ) x Φ table (only for the positive portion) can be found in many text books on Probability. The relation, ( ) 1 ( ) x x Φ − = − Φ , will help you to get ( ) x Φ for negative values of its variable. • Another special function defined as, 2 2 1 ( ) 2 u x Q x e du π +∞ − ∫ ¡ , measuring the areas in the tail portion of the Gaussian pdf, is commonly used in communication systems. • Relation between the two special functions: ( ) 1 ( ) x Q x Φ = − . Important Properties of a Gaussian r.v., (0,1) X N ∼ : { } { } { } ( ) { } ( ) { } { } { } { } { } { } 2 2 2 2 2 2 1 ( ) 0,--- zero-mean 2 1 1,--- unit-variance 2 1 1 1 68.3%68....
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This note was uploaded on 09/14/2011 for the course ECE 321 taught by Professor Hongyage during the Spring '11 term at NJIT.

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Gaussian+pdf+and+special+properties - Gaussian...

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