uniform

Uniformdistribution(discrete)

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Inprobabilitytheoryandstatistics,thediscreteuniform

distributionisasymmetricprobabilitydistributionwherebyafinite

numberofvaluesareequallylikelytobeobrved;everyone

ofnvalueshaqualprobability1/rwayofsaying"discrete

uniformdistribution"wouldbe"aknown,finitenumberofoutcomes

equallylikelytohappen".

Asimpleexampleofthediscreteuniformdistributionisthrowinga

siblevaluesare1,2,3,4,5,6,andeachtime

thediceisthrowntheprobabilityofagivenscoreis1/ice

arethrownandtheirvaluesadded,theresultingdistributionisno

longeruniformsincenotallsumshaveequalprobability.

Thediscreteuniformdistributionitlfisinherentlynon-parametric.

Itisconvenient,however,toreprentitsvaluesgenerallybyan

integerinterval[a,b],sothata,bbecomethemainparametersofthe

distribution(oftenonesimplyconsiderstheinterval[1,n]withthe

singleparametern).Withtheconventions,thecumulative

distributionfunction(CDF)ofthediscreteuniformdistributioncanbe

expresd,foranyk∈[a,b],as

Estimationofmaximum

Thixampleisdescribedbysayingthatasampleofkobrvationsis

obtainedfromauniformdistributionontheintegers,

problemiscommonlyknownastheGermantankproblem,following

theapplicationofmaximumestimationtoestimatesofGermantank

productionduringWorldWarII.

TheUMVUestimatorforthemaximumisgivenby

wheremisthesamplemaximumandkisthesamplesize,sampling

nbeenasaverysimpleca

ofmaximumspacingestimation.

Theformulamaybeunderstoodintuitivelyas:

"Thesamplemaximumplustheaveragegapbetweenobrvationsinthe

sample",

thegapbeingaddedtocompensateforthenegativebiasofthesample

maximumasanestimatorforthepopulationmaximum.

Thishasavarianceof

soastandarddeviationofapproximately,the(population)

averagesizeofagapbetweensamples;compareabove.

Thesamplemaximumisthemaximumlikelihoodestimatorforthe

populationmaximum,but,asdiscusdabove,itisbiad.

Ifsamplesarenotnumberedbutarerecognizableormarkable,one

caninsteadestimatepopulationsizevia

thecapture-recapturemethod.

discreteuniform

Probabilitymassfunction

n=5wheren=b

−

a+1

Cumulativedistributionfunction

Parameters

Support

pmf

CDF

Mean

Median

ModeN/A

Variance

Skewness

Ex.

kurtosis

Entropy

MGF

CF