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DiscreteProbabilityDistributions

Chapter 6Dr.Richard Jerz

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GOALS

• Define the termsprobability distribution andrandom variable.

• Distinguish between discrete andcontinuousprobability distributions.

• Calculate themean,variance,and standarddeviation ofadiscreteprobabilitydistribution.

• Describe the characteristics ofand computeprobabilities usingthe binomial probabilitydistribution.

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WhatisaProbabilityDistribution?

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Experiment: Toss a coin three times. Observe the number of heads. The possible results are: zero heads, one head, twoheads, and three heads. What is the probability distribution for the number of heads?

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ADistribution

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10

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012345678910

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Coun

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#inBag

#ofGreenM&M'sinBag

AProbability Distribution

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5.6%0.0%

50.0%

16.7% 16.7%11.1%

0.0%0%

10%

20%

30%

40%

50%

60%

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Percen

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#inBag

ProbabilityofGreenM&M'sinBag

ProbabilityforDiceToss

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Tossing2Dice

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ProbabilityDistribution,#Heads,3CoinTosses

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CharacteristicsofaProbabilityDistribution

• Theprobability ofaparticular outcome isbetween 0and 1inclusive

• Theoutcomes aremutually exclusive events• Thelist isexhaustive,the sumoftheprobabilities ofthevariousevents isequal to1

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RandomVariables

• Random variable- aquantity resulting fromanexperiment that,bychance, canassumedifferent values.

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TypesofRandomVariables

• Discrete Random Variable canassumeonlycertain clearlyseparated values.Itisusuallytheresult ofcounting something

• Continuous Random Variable canassumeaninfinite numberofvalues within agivenrange.Itisusually theresult ofsometypeofmeasurement

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DiscreteRandomVariablesExamples

• Thenumber ofstudents inaclass.• Thenumber ofchildren inafamily.• Thenumber ofcarsentering acarwash inahour.

• Numberofhomemortgagesapproved byCoastal FederalBanklast week.

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ContinuousRandomVariablesExamples

• Thedistance students traveltoclass.• Thetime ittakesanexecutiveto drivetowork.

• Thelength ofanafternoon nap.• Thelength oftime ofaparticular phone call.

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CharacteristicsofaDiscreteDistribution

• Themain featuresofadiscrete probabilitydistribution are:• Thesumoftheprobabilitiesofthevariousoutcomesis1.00.

• Theprobabilityofaparticularoutcomeisbetween0and1.00.

• Theoutcomesaremutuallyexclusive.

and• theitembeingmeasuredcanassumeonlycertainseparated(counted)values.

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TheMeanofaProbabilityDistribution

• The mean is a typical value used to represent the central location of a probability distribution.

• The mean of a probability distribution is also referred to as its expected value.

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[ ( )]xP xµ =∑

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Similarto“WeightedMean”

• Theweighted meanofasetofnumbersX1,X2,...,Xn,withcorresponding weights w1,w2,...,wn,iscomputed fromthe followingformula:

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1 1 2 2

1 2

n nw

n

w X w X w XXw w w+ + +

=+ + +

KK

TheVarianceandStandardDeviation

• Measurestheamount ofspreadinadistribution

• Thecomputational stepsare:1. Subtractthemeanfromeachvalue,andsquare

thisdifference.2. Multiplyeachsquareddifferencebyits

probability.3. Sumtheresultingproductstoarriveatthe

variance.

• Thestandard deviation isfound bytakingthesquarerootof thevariance.

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2 2[( ) ( )]x u P xσ = −∑Mean,Variance,andStandardDeviationExample

• JohnRagsdalesellsnewcarsforPelicanFord.JohnusuallysellsthelargestnumberofcarsonSaturday.HehasdevelopedthefollowingprobabilitydistributionforthenumberofcarsheexpectstosellonaparticularSaturday.

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MeanofaProbabilityDistributionExample

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VarianceandStandardDeviationExample

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BinomialProbabilityDistribution(discrete)

Characteristics:• Thereareonlytwo possible outcomes on aparticular trial ofanexperiment.

• Theoutcomes aremutually exclusive,• Therandom variableistheresult ofcounts.• Eachtrial isindependent ofanyothertrial• Examples:

• Yesorno• Trueorfalse• Onoroff• Correctorincorrect

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TreeDiagrams

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BinomialTree

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BinomialProbabilityFormula

Where• Cdenotes acombination.• n isthenumber oftrials• xisthe randomvariable defined asthenumber ofsuccesses.

• πisthe probability ofasuccesson eachtrial

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ExamplewithM&M’s

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Brown M&M (Binomial)

0.00

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1.00

0 >0

2 Conditions

Prob

abili

ty

Question:

• What isthe probability that2out of4bagshavebrownM&M’s?• N=4,x=2,π = .86,

• How about 3out of4bagshavingbrownM&M’s?• N=4,x=3,π = .86

• How about between 2and3out of4bagshavingbrownM&M’s?• N=4,x=3?,π = .86

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Example:BinomialProbability

• Therearefiveflightsdaily fromPittsburghviaUSAirways intotheBradfordPennsylvaniaRegionalAirport.Supposetheprobabilitythatany flightarriveslateis.20.

• Whatistheprobabilitythatnoneoftheflightsarelatetoday?

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BinomialDistributionMeanandVariance

• Meanofabinomial distribution

• Variance ofabinomial distribution

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BinomialDist.– MeanandVariance:Example

• Fortheexampleregardingthenumberoflateflights,recallthatπ =.20andn=5.

• Whatistheaveragenumberoflateflights?

• Whatisthevarianceofthenumberoflateflights?

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BinomialDist.- MeanandVariance:AnotherSolution

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BinomialDistribution- Table• Fivepercentofthewormgearsproducedbyanautomatic, high-speedCarter-Bell millingmachine aredefective.What is theprobabilitythat outof sixgears selectedat random nonewillbedefective?Exactly one?Exactly two?Exactly three?Exactly four?Exactly five?Exactlysixoutofsix?

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Binomial – ShapesforVaryingπ(n constant)

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Binomial– ShapesforVaryingnwith(π constant)

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CumulativeBinomialProbabilityDistributions

• AstudyinJune 2003bytheIllinoisDepartment ofTransportation concluded that76.2percentoffront seatoccupants usedseatbelts. Asample of12vehicles isselected.What istheprobability thefrontseatoccupants in atleast7ofthe12vehiclesarewearing seatbelts?

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BinomialProbability- Excel

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CumulativeBinomialProbabilityDist.InExcel

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