Continuous Random Variables Expected Values …

← Back to document page

Continuous Random VariablesExpected Values and MomentsStatistics 110Summer 2006Copyrightc 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= 1Continuous Random Variables1For anya < b, the probability thatP[a < X < b]is the area under thedensity curve (x)abP[a < X < b] = baf(x)dxContinuous Random Variables2Note thatf(a)isNOTthe probability of observingX=aasP[X=a] = aaf(x)dx= 0Thus the probability that a Continuous RV takes on any particular value is0.

Expected Values and Moments Deflnition: The Expected Value of a continuous RV X (with PDF f(x)) is E[X] = Z 1 ¡1 xf(x)dx assuming that R1 ¡1 jxjf(x)dx < 1. The expected value of a distribution is often referred to as the mean of the

  Value, Expected, Variable, Continuous, Random, Expected value, Continuous random variables expected