Transcription of Chapter 3 Continuous Random Variables
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Chapter 3 Continuous Random Variables
Chapter 3 Continuous Random IntroductionRather thansummingprobabilities related to discrete Random Variables , here forcontinuous Random Variables , thedensitycurve isintegratedto determine (Introduction)Patient s number of visits,X, and duration of visit, =value of function,F(3) = P(Y < 3) = 5/12x , pmf f(x)probability ( distribution ): cdf F(x)probability less than = sum of probabilityat specific valuesP(X < ) = P(X = 0) + P(X = 1)= + = (X = 2) = , pdf f(y) = y/6, 2 < y < 4probability less than 3 = area under curve,P(Y < 3) = 5/12xprobability at 3,P(Y = 3) = 0probability less than = value of functionF( ) = P(X < ) = : Comparing discrete and Continuous distributions7374 Chapter 3. Continuous Random Variables (LECTURE NOTES 5)1. Number of visits,Xis a (i)discrete(ii)continuousrandom variable ,and duration of visit,Yis a (i)discrete(ii)continuousrandom (a)P(X= 2) = (i)0(ii) (iii) (iv) (b)P(X ) =P(X 1) =F(1) = + = (i)summation(ii)integrationand is a value of a(i)probability mass function(ii)cumulative distribution functionwhich is a (i)stepwise(ii)smooth increasingfunction(c)E(X) = (i) xf(x)(ii) xf(x)dx(d)V ar(X) = (i)E(X2) 2(ii)E
Random variable Xis continuous if probability density function (pdf) fis continuous at all but a nite number of points and possesses the following properties: f(x) 0, for all x, R 1 1 f(x) dx= 1, P(a<X b) = R b a f(x) dx The (cumulative) distribution function (cdf) for random variable Xis F(x) = P(X x) = Z x 1 f(t) dt; and has properties lim x ...
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