Transcription of Continuous Random Variables Expected Values and Moments
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Continuous Random Variables Expected Values and Moments
Continuous Random VariablesExpected Values and MomentsStatistics 110 Summer 2006 Copyrightc 2006 by Mark E. IrwinContinuous Random VariablesWhen defining a distribution for a Continuous RV, the PMF approach won tquite work since summations only work for a finite or a countably infinitenumber of items. Instead they are based on the followingDefinition:LetXbe a Continuous RV. TheProbability Density Function(PDF) is a functionf(x)on the range ofXthat satisfies the (x) f(x) 0 fis piecewise Continuous f(x)dx= 1 Continuous Random Variables1 For anya < b, the probability thatP[a < X < b]is the area under thedensity curve (x)abP[a < X < b] = baf(x)dxContinuous Random Variables2 Note thatf(a)isNOTthe probability of observingX=aasP[X=a] = aaf(x)dx= 0 Thus the probability that a Continuous RV takes on any particular value is0.
Continuous Random Variables When deflning a distribution for a continuous RV, the PMF approach won’t quite work since summations only work for a flnite or a countably inflnite
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