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GAUSSIAN RANDOM VECTORS AND PROCESSES

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, , ,wellknown, ,theseassumptionsareoftenapproximatelysa tisfied,sotheresults,ifusedwithinsightan dcare, ,butstartswithastudyofGaussian(normal1)r andomvariablesandvectors, (rv)WisdefinedtobeanormalizedGaussianrvi fithasthedensityfW(w)=1p2 exp w22 ;forallw2R.( )1Gaussianrv sareoftencallednormalrv ,firstbecausethecorrespondingprocessesar eusuallycalledGaussian,secondbecauseGaus sianrv s(whichhavearbitrarymeansandvariances)ar eoftennormalizedtozeromeanandunitvarianc e,andthird,becausecallingthemnormalgives thefalseimpressionthatotherrv (w)integratesto1( ,itisaprobabilitydensity), , ,ifweconsiderthervZ= W,thenthedistributionfunctionsofZandWare relatedbyFZ( w)=FW(w).

110 CHAPTER 3. GAUSSIAN RANDOM VECTORS AND PROCESSES Exercise 3.1 shows that f W(w) integrates to 1 (i.e., it is a probability density), and that W has mean 0 and variance 1. If we scale a normalized Gaussian rv W by a positive constant , i.e., if we consider the

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