on 09-Dec-2018 (Sun)

Suchmodelsare probabilitydistributions, which are just functionsdescribinghow the probabilityofeachoutcome depends on the valueofthatoutcome.

n particular, we introducefive specific probabilitymodelsfor discrete data. Threeofthese relatedirectly to Bernoulli trials,which are experimentsresultinginabinary outcome. Thesethree distributionsare theBernoulli distribution, the binomial distributionand the geometric distribution.Wealsointroducean importantprobabilitymodel fordatainthe form of counts, calledthe Poissondistribution.(It hasaless direct connection to Bernoullitrials.) Thefinaldiscretedistribution thatisintroduced is onefor situationswhere no valueinthe range of possiblevalues is morelikelythanany othervalue to occur; this is thediscreteuniform distribution

Theterm Bernoulli trial is use dtodescribeasingle statistical exp eriment forwhich thereare twopossibleoutcomes. So takinganitem from the productionlinetodeterminewhetherornot it is defective is a Bernoulli trial; and each shotbyanarcheratatargetisaBernoullitrial – thearrow either hits or missesthe centreofthe target; andeachtennis match mayberegarded as aBernoulli trial in whichthe tennisplayer either wins or loses;and so on

Asingle Bernoulli trialissosimple asituation that thereisonlyone possibleset of probability distributionsthatcan describe it.Thisiscalled the Bernoulli probabilitydistribution

Whereanexperiment involves aBernoulli trial, it is usualtomatch the number 1toone outcomeand the number 0tothe othe r; then the outcome of aBernoulli trialisarandomvariable with range {0, 1}.

Randomvariables that cantakeonlytwo possiblevaluesare called Bernoulli randomvariables.

Supposethat X is theoutcome of asingle Bernoullitrial, taking the value0or 1. Then X hasaBernoulliprobabilitydistribution, or Bernoulli distribution forshort.Ifwelet p denotethe probabilitythat X takesthe value1,then X is said to have aBernoulli distribution with parameter p.Since P (X =1)+P(X =0)=1, the probabilitymassfunction of X is P (X =1)=p(1)=p and P (X =0)=p(0)=1−p. Here, as usual, p() denotesthe p.m.f., butthe letter p is also used conventionally for the Bernoulli parameter. That is, p(x)= ) 1−px=0 px=1.

llBernoulli distributionshavethisform so there is a family of Bernoulli distributions–the valueofpdeterminesaparticular member of the family. Theparameter p is therebysaidtoindex thefamilyofBernoulli distributions. Statisticians are notespeciallyconsistent in their terminologyhere: either an individualmemberofthe Bernoullifamily of distributions(with aparticular valuechosenfor p)orthe whole familyof Bernoulli distributionscan be referredtoas‘the Bernoulli distribution’

Thep.m.f. of aBernoulli distributionmay alsobewritten in multiplicative form as p(x)=p x (1 − p) 1−x ,x=0,1. As willbecome apparent in Subsection 2.1,thisgives p(0)and p(1)ina single expression

TheBernoulli probability model Adiscrete randomvariable X with range {0, 1} is said to have a Bernoulli distributionwithparameter p,where 0

<1, if it hasprobabilitymassfunction p(x)= ) 1−px=0 px=1 or equivalently p(x)=p x (1 − p) 1−x ,x=0,1. (1) This is written X ∼ Bernoulli(p).Thesymbol‘∼’isread‘is distributed as’orsimply ‘is’. Themodel maybeappliedtothe outcome ofaBernoullitrial where the probabilityofobtainingthe outcome 1isequal to p.

Thebinomial distribution If theprobability of successineachofasetofnindependent Bernoulli trialshas thesame value p,the nthe randomvariable X, whichrepresentsthe totalnumberofsuccessesinthe n trials,issaid to haveabinomialdistribution with parameters n and p.This is written X ∼ B(n, p)

Sothese t hree trials satisfythe conditionsfor the totalnumberof headsinthe three tosses to haveabinomialdistribution with parameters n =3and p =1/2. That is,the probability distribution forthe total number of headsisbinomial, B(3, 1/2)

adults withinahousehold arelikelytoinfluenceeachother in the waytheyvote, so the X scoreswould notbeindependent thusthe random variable Y would not have abinomial distribution

Bythe way, ‘successes’ and‘failures’ are the standard ways of referringto the 1s and0sthatare theoutcomesofacollection of Bernoullitrials.This is so whet he rornot asuccessisreally asuccess! Forinstance, in several examplesinUnit2,weconsideredaBernoulli trialinwhich ‘1’ correspondedtoapersonwho wasnot curedbyamedical treatment, ‘0’ to someonewho wascured;inActivity 3(b), a‘success’ wouldbe‘being obese’. Thesuccess/failure terminologyisjust astandard convenience, ‘success’ beinganame forthe fo cusofwhatisbeing studied;you should alwaysreport outcomesand interpretresultsinlanguage that is meaningful in the context at hand

Numberofcombinationsof objectsoftwo types Thenumberofdifferentwaysofordering x objects of onetypeand n − x objects of asecond type in asequence of n objects is given by ! n x - = n! x!(n−x)! . (2) This formulaholdsfor anyinteger valueofxfrom 0ton.The number ' n x , is read ‘n choose x’.Alternative notation for ' n x , is n C x ,sometimes read as ‘n c x’ The number x!isread‘xfactorial’. Forany positiveinteger x,the notation x!isshorthand forthe number 1 × 2 × 3 ×···×x.The number 0! is definedtobe1

Thequantities ' n x , are often called binomialcoefficients.

Thebinomial probability model If arandomvariable X hasabinomialdistribution with parameters n and p,where 0

<1, then it hasprobabilitymassfunction p(x)= ! n x - p x (1 − p) n−x ,x=0,1,2,...,n. (3) This is written X ∼ B(n, p). Thebinomial distribution provides aprobabilitymodel forthe total number of successe sinasequence of n independent Bernoullitrials, in whichthe probabilityofsuccessinasingle trial is p.

An assumption of thebinomialmodel is that of independencefrom trial to trial

Screencast 3.1The shapes of binomial distributions

The range ofAlso, individualbinomial probabilitiesare non-negative. the binomial distribution is {0, 1, 2,...,n} so the summation in question is $ n x=0 p(x). Thekey to showingthat $ n x=0 p(x)=1, as youwill seeinthe next activity,isthe binomial theorem of mathematics, which canbewritten (a + b) n = n 1 x=0 ! n x - a x b n−x .

In Section 1, aBernoulli trialwas definedtobeastatisticalexperiment in whichexactlyone of twopossibleoutcomesoccurs.The outcomesmay be, forinstance, Success–Failure,Yes–No, On–Off or Male–Female. It is usual toidentifyoneofthe outcomes(success) with the number 1and the other (failure) with the number 0. If p is the probabilitythatasingle trial results in asuccess, thenthe outcome of asingle trial is arandomvariable whichhas aBernoulli distribution with parameter p (0

<1).Also, the totalnumberofsuccessesinasequence of n independent trials,eachwith the same probability of success p,isarandomvariable thathas abinomial distribution with parameters n and p,where n is the totalnumberoftrials.

In this section anotherprobabilitymodel associated with asequence of independent Bern oullitrials is introduced;thisisamode lfor arandom variablewhich represents the number of trials up toand including the first success. Thismodel –the geometricdistribution

Thegeometricdistribution Supposethatinasequence of inde pendent Bernoullitrials, the probabilityofsuccessisconst antfromtrial to trialand equal to p, where0

<1. Then thenumberoftrialsuptoand including the first successisarandomvariable X with probability massfunction given by p(x)=P(X=x)=(1 − p) x−1 p,x=1,2,3,.... (4) Therandomvariable X is said to haveageometricdistribution with parameter p.This is written X ∼ G(p).

Mathematically,the probabilities P (X =1), P (X =2), P (X =3), ... form ageometric progression:eachprobabilityinthe sequence is a constant multiple (inthiscase1−p)ofthe precedingone.Thisisthe reason forthe name ‘geometric’. TheAncient Greek mathematicianEuclid knew about geometric progressions Moreover, since0

<1, it follows that 0 < 1 − p<1also, so each consecutive probabilityissmallerthanthe precedingone.Inother words, the geometric p.m.f.isalwaysadecreasing function of x

In Figure 2(a),the parameter p
is equal to 0.8–thatisquite high:you wouldbeunlikelytohavetowait
longfor thefirstsuccessful trial. In Figure 2(b),the valueofthe parameter
p is much lower–the probabilityofsuccessisonly0.3. In thiscase, you
mighthavetowaitquite alongtime forthe first successtooccur.
Whatever thevalue of p,however,the decreasing natureofthe p.m.f.
meansthatthe most likelyvalueofXis 1.

there isnoconvenientformulafor the c.d.f.ofabinomialrandomvariable.However,itispossibletofind a simple formulafor the c.d.f.ofageometric randomvariable X.Moreover, the methodfor finding thissimple formulaissimple tooifweuse alittle trick: first find an expression forthe probability P (X>x)and thenobtain an expression forthe c.d.f. F (x)=P(X≤x)=1−P(X>x). Now, P (X>x)isthe probabilitythatmore than x trials areneeded to obtain asuccess, or equivalently,thatall thefirst x trials resultinfailure. This probabilityisequal to (1 − p) x .Itfollowsthat F (x)=1−P(X>x)=1−(1 − p) x . This is the formulaweare seeking forthe c.d.f. of ageometric distribution, stated in thebox below. Thec.d.f.ofageometric distribution If therandomvariable X hasageometric distribution with parameter p,thenthe c.d.f. of X is given by F (x)=P(X≤x)=1−(1 − p) x , (5) for x =1,2,3,....