Session 2: Probability distributions and density …

1 Session2: p. :Probabilitydistributionsanddensityfunct ions p. :Probabilitydistributionsanddensityfunct ions p. 3 RandomvariablesSession2:Probabilitydistr ibutionsanddensityfunctions p. 4 RandomVariablesA randomvariableis ,thevalueofthefirstrollofa ,thesumofa rolloftwo ,therollofa dice,ortheoutcomeofahorserace, ,thenumberofwhitehaironmyhead,orhow muchdividendINFOSYSTCH willannouncenextyear, :ContinuousRVscanhave a fixedminimumormaximum,however, :Probabilitydistributionsanddensityfunct ions p. 5 ProbabilitydistributionsSession2:Probabi litydistributionsanddensityfunctions p. 6 Whatis a probabilitydistribution?Fora discreteRV, theprobabilitydistribution(PD)isa ,intherollofa die:ValueProbabilityValueProbability11/6 41/621/651/631/661/6A probabilitydistributionwillcontainallthe outcomesandtheirrelatedprobabilities, :Probabilitydistributionsanddensityfunct ions p. 7 Howtoreada probabilitydistribu-tionFromthedistribut ion,wecanfind:X = 3Pr(X= 3) = 1=6X = evennumberPr(X= 2orX= 4orX= 6) = 3=6 = 1=2 Moreinteresting,wecanalsofind:Pr(X >3) = 3=6 = 1=2 Thisis calledacumulative :Probabilitydistributionsanddensityfunct ions p.

2 8 Whatis a cumulative probabilitydis-tribution(CD)?A 11/6X 44/6X 22/6X 55/6X 33/6X 61 TheCDis a monotonicallyincreasingsetofnumbersTheCD alwaysendswithat :Probabilitydistributionsanddensityfunct ions p. 9 Examplesofa PD:BernoulliRVsBernoullidistribution:The outcomeis eithera failure (0)ora success (1).Xis a bernoulliRV whenPr(X= 0)=pPr(X= 1)=(1 p)Forexample,theUSD-INRrisesornotat , :Probabilitydistributionsanddensityfunct ions p. 10 TheCDofa BernoulliRVWe needtoknow whatpoftheRV 0 is:ValueProbabilityX :Probabilitydistributionsanddensityfunct ions p. 11 Examplesofa PD:BinomialRVsBinomialdistribution:Theou tcomeis a sum(s) ofa setofbernoullioutcomes(n).Forexample,the numberoftimesUSD-INRroseinthelast10days? Pr(X=s) =n!(n s)!s!ps(1 p)(n s)Here,pis theprobabilitythattheUSD-INRroseinaday. Theassumptionis thatpis :p, :Probabilitydistributionsanddensityfunct ions p. 12 Explainingn!n!is mathematicalshort-handfortheproduct1 2 3 4 : : :(n 2) (n 1) nExample:5!

3 = 1 2 3 4 5 Note:0! = 1! = 1 Session2:Probabilitydistributionsanddens ityfunctions p. 13 Testingconcepts:BinomialRV CDA bernoullieventhasa theCDofa binomialRV thatis thenumberoftimesthatsuccesscanbeacheived in4 trials? :Probabilitydistributionsanddensityfunct ions p. 14 Testingconcepts:BinomialRV CDA bernoullieventhasa theCDofa binomialRV thatis thenumberoftimesthatsuccesscanbeacheived in4 trials? :Probabilitydistributionsanddensityfunct ions p. 14 Testingconcepts:A discreteRVA discreteRV hasthefollowingPD:X12458Pr(X) (4). ((Pr(x)= 2)or(Pr(x)= 4)). (x 4). (x <4). :Probabilitydistributionsanddensityfunct ions p. 15 Testingconcepts:A discreteRVA discreteRV hasthefollowingPD:X12458Pr(X) (4). ((Pr(x)= 2)or(Pr(x)= 4)). (x 4). (x <4). :Probabilitydistributionsanddensityfunct ions p. 15 ProbabilitydensityfunctionsSession2:Prob abilitydistributionsanddensityfunctions p. 16 Whatisa probabilitydensityfunc-tion?Theprobabili tydensityfunction(PDF)is thePDofa ,it is ,thePDFis alwaysa functionwhichgivestheprobabilityofoneeve nt, wedenotethePDFasfunctionf, thenPr(X=x) =f(x)A probabilitydistributionwillcontainallthe outcomesandtheirrelatedprobabilities, :Probabilitydistributionsanddensityfunct ions p.

4 17 TheproblemwithestimatingPDFsIna setofcontinuousrandomvariables,theprobab ilityofpickingouta definethePr(X= x)asthefollowingdifference:Pr(X (x+ )) Pr(X x)as ,Pr(X x) is thecumulative :Probabilitydistributionsanddensityfunct ions p. 18 Whatis thecumulative densityfunc-tion(CDF)?Analogoustothedisc reteRV case,theCDFis thecumulationoftheprobabilityofalltheout comesuptoa , theCDFis theprobabilitythattheRV cantakeany weassumethattheRVXcantake valuesfrom 1to1, thentheoretically,F(X) =ZX 1f(x)d(x)Session2:Probabilitydistributio nsanddensityfunctions p. 19 ReformulatingthePDFincalculusPr(x= X)is givenas:Pr(x=X) =F(X+ ) F(X)=d(F(x))=d(x)forinfinitesimallysmall .ForcontinuousRVs,weapproachthePr(x)asth ederivative :Probabilitydistributionsanddensityfunct ions p. 20 Examplesofa PDF:UniformRVsUniformdistribution:Theout comeis any numberthatcantake a valuebetweena minimum(A) andamaximum(B) uniformRV,Pr(X=x) = 1=(B A)Theuniformdensityhastwo parameters,A; :Probabilitydistributionsanddensityfunct ions p.

5 21 Testingconcepts:UniformdensityCDWhatis theformoftheUniformCDF, giventhatthemaximum=Bandminimum=A?F(X) =ZXminimumf(x)d(x)=ZXA1=(B A)d(x)=(X A)=(B A)Session2:Probabilitydistributionsandde nsityfunctions p. 22 Testingconcepts:UniformdensityCDWhatis theformoftheUniformCDF, giventhatthemaximum=Bandminimum=A?F(X) =ZXminimumf(x)d(x)=ZXA1=(B A)d(x)=(X A)=(B A)Session2:Probabilitydistributionsandde nsityfunctions p. 22 ExamplesofaPDF:Nor-mal/GaussianRVf(x) =1p2 e 12((x )= )2 Thenormalhastwoparameters, ; .A :Probabilitydistributionsanddensityfunct ions p. 23 FeaturesofthenormalPDFRV scantake valuesfrom is symmetric:Pr( x) = Pr(x)WhenXis a normalRV withparameters ; , thenY= 5:6 + 0:2 Xwillalsobea normalRV, withknownparameters(5:6 + 0:2 );(0:2 ).Note:Specialcaseofa normaldistributionis = 0; = 1. Thisis :Probabilitydistributionsanddensityfunct ions p. 24 Probabilitydistributions 4 densityNormal probabilitySession2:Probabilitydistribut ionsanddensityfunctions p.

6 25 ProblemstobesolvedSession2:Probabilitydi stributionsanddensityfunctions p. 26Q1:BinomialRVsA door-to-doorsalespersonhasfoundthathersu ccessrateinsellingis thesalespersoncontactsthreepersons,whati s leastone?Session2:Probabilitydistributio nsanddensityfunctions p. 27Q2:UniformdensityCDIfB= 10andA= Pr(x= )? Pr(x= )? Pr(x> 7)?Session2:Probabilitydistributionsandd ensityfunctions p. 28Q3:Session2:Probabilitydistributionsan ddensityfunctions p. 29 ReferencesChapter2, ,2001,7theditionSession2:Probabilitydist ributionsanddensityfunctions p. 30