Gaussian Random Variables and Processes

Gaussian Random Variables and ProcessesSaravanan of electrical EngineeringIndian Institute of Technology BombayAugust 1, 20121 / 33 Gaussian Random VariablesGaussian Random VariableDefinitionA continuous Random variable with pdf of the formp(x) =1 2 2exp( (x )22 2), <x< ,where is the mean and 2is the variance. 4 2024 (x)3 / 33 Notation N( , 2)denotes a Gaussian distribution with mean andvariance 2 X N( , 2) Xis a Gaussian RV with mean andvariance 2 X N(0,1)is termed a standard Gaussian RV4 / 33 Affine Transformations Preserve GaussianityTheoremIf X is Gaussian , then aX+b is Gaussianfor a,b IfX N( , 2), thenaX+b N(a +b,a2 2). IfX N( , 2), thenX N(0,1).5 / 33 CDF and CCDF of Standard Gaussian Cumulative distribution function (x) =P[N(0,1) x] = x 1 2 exp( t22)dt Complementary cumulative distribution functionQ(x) =P[N(0,1)>x] = x1 2 exp( t22)dtxtp(t)Q(x) (x)6 / 33 Properties ofQ(x) (x) +Q(x) =1 Q( x) = (x) =1 Q(x) Q(0) =12 Q( ) =0 Q( ) =1 X N( , 2)P[X> ] =Q( )P[X< ] =Q( )7 / 33 Jointly Gaussian Random VariablesDefinition (Jointly Gaussian RVs) Random variablesX1,X2.

Gaussian Random Variables and Processes Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 1, 2012 1/33. Gaussian Random Variables. Gaussian Random Variable Definition A continuous random variable with pdf of the form

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