Continuous Random Variables: The Uniform Distribution

1 Connexionsmo dule:m168191 ContinuousRandomVariables:TheUniformDist ribution SusanDeanBarbaraIllowsky, ducedbyTheConnexionsPro jectandlicensedundertheCreativeCommonsAt tributionLicense AbstractThismo duledescrib estheprop ertiesoftheUniformDistributionwhichdescr ib ,inseconds, ,inseconds,followauniformdistributionb etween0and23seconds, ,inseconds,ofaneight-weekoldbaby' U(a,b)wherea=thelowestvalueofXandb= (X) =1b afora X ,X U(0,23)andf(X) =123 0for0 X :Feb20,20097:18pmUS/Central URL: Saylor URL: Attributed to: Barbara Illowsky and Susan Dean Page 1 of 6 Connexionsmo dule:m168192 =a+b2and = (b a)212 Forthisproblem,thetheoreticalmeanandstan darddeviationare =0+232= = (23 0)212= etween2and18seconds?

2 SolutionFindP(2< X <18).P(2< X <18) = (base) (height) = (18 2) 123= ercentileforaneightweekoldbaby' ercentofthesmilingtimesfallb elowthe90thp ercentile,k,soP(X < k) = (X < k) = (base) (height) = (k 0) 123= 23 = (X >12|X >8) rstway, URL: Saylor URL: Attributed to: Barbara Illowsky and Susan Dean Page 2 of 6 Connexionsmo dule:m168193 Writeanewf(X):f(X) =123 8=115for8< X <23P(X >12|X >8) = (23 12) 115=1115 Forthesecondway,usetheconditionalformula fromProbabilityTopicswiththeoriginaldist ributionX U(0,23):P(A|B) =P(AANDB)P(B)Forthisproblem,Ais(X >12)andBis(X >8).

3 So,P(X >12|X >8) =(X>12 ANDX>8)P(X>8)=P(X>12)P(X>8)=11231523= :Theamountoftime,inminutes,thatap ersonmustwaitforabusisuniformlydistribut edb etween0and15minutes, erofminutesap U(0,15). (X) =115 0=115for0 X (X < ). (X < k) = (base) (height) = ( 0) 115= URL: Saylor URL: Attributed to: Barbara Illowsky and Susan Dean Page 3 of 6 Connexionsmo dule:m168194 Problem2 Ontheaverage,howlongmustap ersonwait?Findthemean, ,andthestandarddeviation, .Solution =a+b2=15+02= ,ap = (b a)212= (15 0)212= ercentofthetime,thetimeap ersonmustwaitfallsb elowwhatvalue?

4 Note:Thisasksforthe90thp (X < k) = (base) (height) = (k 0) (115) =k 115k = ( ) (15) = ercentofthetime,ap :Theaveragenumb erofdonutsanine-yearoldchildeatsp , erofdonutsanine-yearoldchildeatsp U( ,4).Problem1( ) URL: Saylor URL: Attributed to: Barbara Illowsky and Susan Dean Page 4 of 6 Connexionsmo dule:m168195 Problem2( )Findtheprobabilitythatadi (referto"ProbabilityTopics1").Youareaskedto ndtheprobabilitythatanine- erentways(seethe rstexample(Example1)). ,youarenolongerstartingata = (X):f(X) =14 X (X >2|X > ).

5 (X >2|X > ) = (base) (newheight) = (4 2) (2/5) =?Theprobabilitythatanine-yearoldchildea tsanaverageofmorethan2 :DrawtheoriginalgraphforX U( ,4).UsetheconditionalformulaP(X >2|X > ) =P(X>2 ANDX> )P(X> )=P(X>2)P(X> )= =45note:See"SummaryoftheUniformandExp onentialProbabilityDistributions2" "ProbabilityTopics:Intro duction"< >2"ContinuousRandomVariables:SummaryofThe UniformandExp onentialProbabilityDistributions"< > URL: Saylor URL: Attributed to: Barbara Illowsky and Susan Dean Page 5 of 6 Connexionsmo dule:m168196 SolutionstoExercisesinthisMo duleSolutiontoExample4,Problem1( ) ,Problem2( )45 GlossaryDe nition1.

6 ConditionalProbabilityThelikeliho o dthataneventwillo ccurgiventhatanothereventhasalreadyo nition2:UniformDistributionAcontinuousra ndomvariable(RV)thathasequallylikelyoutc omesoverthedomain,a < x < ecausethegraphofthep :X~U(a,b).Themeanis =a+b2andthestandarddeviationis = (b a)212 Theprobabilitydensityfunctionisf(X) =1b afora X (X x) =x ab URL: Saylor URL: Attributed to: Barbara Illowsky and Susan Dean Page 6 of 6