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Bernoulli Distribution - University of Chicago

BernoulliDistributionExample:Tossof coinDe neX= 1 if headcomesupandX= 0 if : (X= 1) = (X= 0) =12 Examples:Often:Two outcomeswhich arenotequallylikely: Successof medicaltreatment Interviewed personis female Student passesexam Transmittanceof a diseaseBernoullidistribution(withparamet er ) Xtakes two values,0 and1, withprobabilitiespand1 p FrequencyfunctionofXp(x) = x(1 )1 xforx2f0;1g0otherwise Often:X= 1if eventAhasoccured0otherwiseExample:A= blood pressureabove 140 , Jan 30, 2003- 1 -BernoulliDistributionLetX1; : : : ; Xnbe independent Bernoullirandomvariableswithsameparamete r.

Classical example from von Bortkiewicz (1898). – Number of fatalities resulting from being kicked by a horse – 200 observations (10 corps over a period of 20 years) Statistical model: – Each soldier is kicked to death by a horse with probability µ. – Let Y be the number of such fatalities in one corps. Then Y » Bin(n;µ)

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Transcription of Bernoulli Distribution - University of Chicago

1 BernoulliDistributionExample:Tossof coinDe neX= 1 if headcomesupandX= 0 if : (X= 1) = (X= 0) =12 Examples:Often:Two outcomeswhich arenotequallylikely: Successof medicaltreatment Interviewed personis female Student passesexam Transmittanceof a diseaseBernoullidistribution(withparamet er ) Xtakes two values,0 and1, withprobabilitiespand1 p FrequencyfunctionofXp(x) = x(1 )1 xforx2f0;1g0otherwise Often:X= 1if eventAhasoccured0otherwiseExample:A= blood pressureabove 140 , Jan 30, 2003- 1 -BernoulliDistributionLetX1; : : : ; Xnbe independent Bernoullirandomvariableswithsameparamete r.

2 FrequencyfunctionofX1; : : : ; Xnp(x1; : : : ; xn) =p(x1) p(xn) = x1+:::+xn(1 )n x1 ::: xnforxi2f0;1gandi= 1; : : : ; nExample:Paired-SampleSignTest Studysuccessof newelaboratesafety program Recordaverageweeklylossesin hoursof labor dueto accidents beforeandafterinstallationof theprogramin 10industrialplantsPlant12345678910 Before457346124335783342617 After366044119355177292411De nefortheith plantXi= 1if rstvalueis greaterthanthesecond0otherwiseResult:111 1011111 TheXi's areindependentlyBernoulli distributedwithunknownparameter .Distributions, Jan 30, 2003- 2 -BinomialDistributionLetX1; : : : ; Xnbe independent Bernoullirandomvariables Oftenonlyinterestedin number of successesY=X1+: : :+XnExample:PairedSampleSignTest(contd)D e nefortheith plantXi= 1if rstvalueis greaterthanthesecond0otherwiseY=nPi=1 XiYis thenumber of plants forwhich thenumber of losthourshasdecreasedaftertheinstallatio nof thesafety programWeknow: Xiis Bernoullidistributedwithparameter Xi's areindependentWhatis thedistributionofY?

3 probability of realizationx1; : : : ; xnwithysuccesses:p(x1; : : : ; xn) = y(1 )n y Number of di erent realizationswithysuccesses: ny Distributions, Jan 30, 2003- 3 -BinomialDistributionBinomialdistributio n(withparametersnand )LetX1; : : : ; Xnbe independent andBernoullidistributedwithpa-rameter andY=nPi=1Xi:Yhasfrequencyfunctionp(y) = ny y(1 )n yfory2f0; : : : ; ngYisbinomially distributedwithparametersnand . We writeY Bin(n; ):Notethat thenumber of trialsis xed, theprobability of successis thesameforeach trial,and :PairedSampleSignTest(contd)LetYbe thenumber of plants forwhich thenumber of losthourshasdecreasedaftertheinstallatio nof thesafety Bin(n; )Distributions, Jan 30, 2003- 4 -BinomialDistributionBinomialdistributio nforn= 10p(x) = (x) = (x) = (x) = , Jan 30, 2003- 5 -GeometricDistributionConsidera sequenceof independent Bernoullitrials.

4 Oneach trial,a successoccurswithprobability . LetXbe thenumber of trialsupto the thedistributionofX? probability of nosuccessinx 1 trials:(1 )x 1 probability of onesuccessin thexth trial: ThefrequencyfunctionofXisp(x) = (1 )x 1;x= 1;2;3; : : :Xisgeometrically distributedwithparameter .Example:Supposea batterhasprobability13to thechancethathemissestheballlessthan3 times?ThenumberXof ballsupto the rstsuccessis geometricallydistributedwithparameter13. Thus (X 3) =13+13 23+13 23 2= 0:7037:Distributions, Jan 30, 2003- 6 -HypergemetricDistributionExample:Qualit y ControlQuality control- sampleandexaminefractionof producedunits Nproducedunits Mdefective units nsampledunitsWhatis theprobability thatthesamplecontainsxdefective units?

5 ThefrequencyfunctionofXisp(x) = Mx N Mn x Nn ;x= 0;1; : : : ; n:Xis ahypergeometricrandomvariablewithparamet ersN,M, :Supposethatof 100applicants fora job50 werewomenand50 weremen,allequallyquali we select10applicants atrandomwhatis theprobability thatxof themarefemale?Thenumber of chosenfemaleapplicants is hypergeometricallydistributedwithparamet ers100,50, (x) = 50x 5010 x 10010 forx2f0; : : : ; ngforx= 0;1; : : : ; , Jan 30, 2003- 7 -PoissonDistributionOftenwe areinterestedin thenumber of events which occurin aspeci cperiod of timeor in a speci careaof volume: Number of alphaparticlesemittedfroma radioactive sourceduringagivenperiod of time Number of telephonecallscominginto anexchangeduringoneunitoftime Number of diseasedtreesper acreof a certainwoodland Number of deathclaimsreceived per day by aninsurancecompanyCharacteristicsLetXbe thenumber of timesa certainevent occursduringa givenunitof time(orin a givenarea,etc).

6 Theprobability thattheevent occursin a given unitof timeisthesameforalltheunits. Thenumber of events thatoccurin oneunitof timeis inde-pendent of thenumber of events in otherunits. Themean(orexpected)rateis .ThenXis aPoissonrandomvariablewithparameter andfrequencyfunctionp(x) = xx!e ;x= 0;1;2; : : :Distributions, Jan 30, 2003- 8 -PoissonApproximationThePoissondistribut ionis oftenusedas anapproximationforbinomialprobabilitiesw hennis largeand is small:p(x) = nx x(1 )n x xx!e with =n .Example:Fatalitiesin PrussiancavalryClassicalexamplefromvonBo rtkiewicz(1898).

7 Number of fatalitiesresultingfrombeingkicked by a horse 200observations(10corpsover a period of 20years)Statistical model: Each soldieris kicked to deathby a horsewithprobability . LetYbe thenumber of such fatalitiesin Bin(n; )wherenis thenumber of soldiersin :Thedataarewellapproximatedby a Poissondistributionwith = 0:61 Deathsper , Jan 30, 2003- 9 -PoissonApproximationPoissonapproximatio nof Bin(40; )p(x) (x) (x) (x) (x) (x) (x) (x) =14 =18 =140 =1400 = 10 = 5 = 1 =110 Distributions, Jan 30, 2003- 10 -ContinuousDistributionsUniformdistribut ionU(0; )Range(0;1)f(x) =1 1(0; )(x) (X) = 2var(X) = 212 ExponentialdistributionExp( )Range[0;1)f(x) = exp( x)1[0;1)(x) (X) =1 var(X) =1 2 NormaldistributionN(.]]

8 2)Range f(x) =1p2 2exp 12 2(x )2 (X) = var(X) = 2 XFrequency 2 101234010203040U(0, )XFrequency 2 101234010203040 Exp( )XFrequency 2 101234010203040N( , 2) 20246U(0, )Exp( )N( , 2)Distributions, Jan 30, 2003- 11.

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