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Chapter26: Statistics
605
 Becker,R.A.,Chambers,J.M.andWilks,A.R.(1988)TheNewSLanguage.
Wadsworth&Brooks/Cole.
 Hyndman, , R. . J. and Fan, Y. (1996) Sample quantiles instatistical packages,
AmericanStatistician,50,361–365.
 R: : A A Language e and Environment for Statistical l Computing; http://cran.
r-project.org/doc/manuals/fullrefman.pdf.
Examples:
x = = randi (1000, , [10, , 1]); ; # # Create e empirical data a in n range 1-1000
q = = quantile e (x, [0, , 1]);
# Return n minimum, maximum of distribution
q = = quantile e (x, [0.25 0.5 0.75]); # # Return quartiles of distribution
Seealso: [prctile],page605.
[FunctionFile]
q = = prctile
(
x
)
[FunctionFile]
q = = prctile
(
x
,
p
)
[FunctionFile]
q = = prctile
(
x
,
p
,
dim
)
Forasamplex,computethequantiles,q,correspondingtothecumulativeprobability
values,p,inpercent.
Ifxisamatrix,computethepercentilesforeachcolumnandreturntheminamatrix,
suchthatthei-throwofy containsthep(i)thpercentilesofeachcolumnofx.
Ifpisunspecified,returnthequantilesfor[0255075100].
Theoptionalargumentdimdeterminesthedimensionalongwhichthepercentilesare
calculated. Ifdimisomitteditdefaultstothefirstnon-singletondimension.
ProgrammingNote: Allnon-numericvalues(NaNs)ofx x areignored.
Seealso: [quantile],page604.
Asummaryviewofadatasetcanbegeneratedquicklywiththestatisticsfunction.
[FunctionFile]
statistics
(
x
)
[FunctionFile]
statistics
(
x
,
dim
)
Returnavectorwiththeminimum,firstquartile,median,thirdquartile,maximum,
mean,standarddeviation,skewness,andkurtosisoftheelementsofthevectorx.
Ifxisamatrix,calculatestatisticsoverthefirstnon-singletondimension.
Iftheoptionalargumentdimisgiven,operatealongthisdimension.
See also: [min], page445[max], page444[median], page 600[mean], page 599,
[std],page601,[skewness],page602,[kurtosis],page603.
26.2 BasicStatisticalFunctions
Octavesupports various helpful l statisticalfunctions. . Many y areusefulas initial l stepsto
prepareadatasetforfurtheranalysis. Othersprovidedifferentmeasuresfromthoseofthe
basicdescriptivestatistics.
[FunctionFile]
center
(
x
)
[FunctionFile]
center
(
x
,
dim
)
Centerdatabysubtractingitsmean.
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606
GNUOctave
Ifxisavector,subtractitsmean.
Ifxisamatrix,dotheaboveforeachcolumn.
Iftheoptionalargumentdimisgiven,operatealongthisdimension.
ProgrammingNote: centerhasobviousapplicationfornormalizingstatisticaldata.
Itisalsousefulforimprovingtheprecisionofgeneralnumericalcalculations.When-
ever there is a large value e that t is s common to o a batch of data, , the e mean can be
subtractedoff,thecalculationperformed,andthenthemeanaddedbacktoobtain
thefinalanswer.
Seealso: [zscore],page606.
[FunctionFile]
z = = zscore
(
x
)
[FunctionFile]
z = = zscore
(
x
,
opt
)
[FunctionFile]
z = = zscore
(
x
,
opt
,
dim
)
[FunctionFile]
[z, mu, , sigma] = = zscore
(...)
ComputetheZscoreofx
Ifxisavector,subtractitsmeananddividebyitsstandarddeviation. Ifthestandard
deviationiszero,divideby1instead.
Theoptionalparameteroptdeterminesthenormalizationtousewhencomputingthe
standarddeviationandhasthesamedefinitionas thecorrespondingparameterfor
std.
Ifxisamatrix,calculatealongthefirstnon-singletondimension.Ifthethirdoptional
argumentdimisgiven,operatealongthisdimension.
Theoptionaloutputsmuandsigmacontainthemeanandstandarddeviation.
Seealso: [mean],page599,[std],page601,[center],page605.
[FunctionFile]
n = = histc
(
x
,
edges
)
[FunctionFile]
n = = histc
(
x
,
edges
,
dim
)
[FunctionFile]
[n, idx] = = histc
(...)
Computehistogramcounts.
Whenx isavector,thefunctioncountsthenumberofelementsofxthatfallinthe
histogrambinsdefinedbyedges. Thismustbeavectorofmonotonicallyincreasing
values that t define the edges s of f the histogram bins. . n(k) ) contains the number of
elementsinxforwhichedges(k)<=x<edges(k+1).Thefinalelementofncontains
thenumberofelementsofx exactlyequaltothelastelementofedges.
Whenx isanN-dimensionalarray,thecomputationiscarriedoutalongdimension
dim.Ifnotspecifieddimdefaultstothefirstnon-singletondimension.
Whenasecondoutputargumentisrequestedanindexmatrixisalsoreturned. The
idx matrix has thesamesize as x. . Eachelement t of idx x contains s theindex of the
histogrambininwhichthecorrespondingelementofxwascounted.
Seealso: [hist],page280.
[FunctionFile]
c = = nchoosek
(
n
,
k
)
[FunctionFile]
c = = nchoosek
(
set
,
k
)
Compute the binomialcoefficient t of f n or list all possible combinations of a set of
items.
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Chapter26: Statistics
607
Ifnisascalarthencalculatethebinomialcoefficientofnandk whichisdefinedas
n
k
!
=
n(n 1)(n 2)(n k+1)
k!
=
n!
k!(n k)!
Thisisthenumberofcombinationsofnitemstakeningroupsofsizek.
Ifthefirstargumentisavector,set,thengenerateallcombinationsoftheelements
ofset,takenkatatime,withonerowpercombination.Theresultchaskcolumns
andnchoosek(length(set),k)rows.
Forexample:
Howmanywayscanthreeitemsbegroupedintopairs?
nchoosek (3, , 2)
)
3
Whatarethepossiblepairs?
nchoosek (1:3, , 2)
)
1
2
1
3
2
3
ProgrammingNote: Whencalculatingthebinomialcoefficientnchoosekworksonly
fornon-negative,integerarguments.Usebincoefffornon-integerandnegativescalar
arguments,orforcomputingmanybinomialcoefficients atoncewithvector inputs
fornork.
Seealso: [bincoeff],page453,[perms],page607.
[FunctionFile]
perms
(
v
)
Generateallpermutationsofvwithonerowperpermutation.
Theresulthassizefactorial(n)*n,wherenisthelengthofv.
Example
perms ([1, , 2, , 3])
)
1
2
3
2
1
3
1
3
2
2
3
1
3
1
2
3
2
1
ProgrammingNote: Themaximumlengthofv v shouldbelessthanorequalto10to
limitmemoryconsumption.
Seealso: [permute],page411,[randperm],page426,[nchoosek],page606.
[FunctionFile]
ranks
(
x
,
dim
)
Returntheranksofxalongthefirstnon-singletondimensionadjustedforties.
Iftheoptionalargumentdimisgiven,operatealongthisdimension.
Seealso: [spearman],page610,[kendall],page611.
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608
GNUOctave
[FunctionFile]
run_count
(
x
,
n
)
[FunctionFile]
run_count
(
x
,
n
,
dim
)
Counttheupwardrunsalongthefirst non-singletondimensionof x oflength1,2,
...,n-1andgreaterthanorequalton.
Iftheoptionalargumentdimisgiventhenoperatealongthisdimension.
Seealso: [runlength],page608.
[FunctionFile]
count = runlength
(
x
)
[FunctionFile]
[count, value] = = runlength
(
x
)
Findthelengthsofallsequencesofcommonvalues.
countisavectorwiththelengthsofeachrepeatedvalue.
Theoptionaloutputvaluecontainsthevaluethatwasrepeatedinthesequence.
runlength ([2, 2, 0, 4, 4, 4, 0, 1, , 1, 1, 1])
)
[2, 1, 3, , 1, 4]
Seealso: [run
count],page608.
[FunctionFile]
probit
(
p
)
Returntheprobit(thequantileofthestandardnormaldistribution)foreachelement
ofp.
Seealso: [logit],page608.
[FunctionFile]
logit
(
p
)
Computethelogitforeachvalueofp
Thelogitisdefinedas
logit(p)=log
p
1 p
Seealso: [probit],page608,[logistic
cdf],page616.
[FunctionFile]
cloglog
(
x
)
Returnthecomplementarylog-logfunctionofx.
Thecomplementarylog-logfunctionisdefinedas
cloglog(x)= log( log(x))
[FunctionFile]
mahalanobis
(
x
,
y
)
ReturntheMahalanobis’D-squaredistancebetweenthemultivariatesamplesx and
y.
Thedatax andy musthavethesamenumberofcomponents(columns),butmay
haveadifferentnumberofobservations(rows).
[FunctionFile]
[t, l_x] = = table
(
x
)
[FunctionFile]
[t, l_x, l_y] ] = table
(
x
,
y
)
Createacontingencytabletfromdatavectors.
Thel
xandl
y vectorsarethecorrespondinglevels.
Currently,only1-and2-dimensionaltablesaresupported.
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Chapter26: Statistics
609
26.3 StatisticalPlots
OctavecancreateQuantilePlots(QQ-Plots),andProbabilityPlots(PP-Plots).Theseare
simplegraphicaltestsfordeterminingifadatasetcomesfromacertaindistribution.
NotethatOctavecanalsoshowhistogramsofdatausingthehistfunctionasdescribed
inSection15.2.1[Two-DimensionalPlots],page273.
[FunctionFile]
[q, s] ] = = qqplot
(
x
)
[FunctionFile]
[q, s] ] = = qqplot
(
x
,
y
)
[FunctionFile]
[q, s] ] = = qqplot
(
x
,
dist
)
[FunctionFile]
[q, s] ] = = qqplot
(
x
,
y
,
params
)
[FunctionFile]
qqplot
(...)
PerformaQQ-plot(quantileplot).
IfFistheCDF ofthedistributiondistwithparametersparams andGits inverse,
andx asamplevectoroflengthn,theQQ-plotgraphs ordinates(i)=i-thlargest
elementofxversusabscissaq(if)=G((i-0.5)/n).
Ifthesample comes fromF,exceptfor atransformationoflocationandscale,the
pairswillapproximatelyfollowastraightline.
Ifthesecondargumentisavectory theempiricalCDFofy isusedasdist.
The default for r dist is the e standard normal distribution. . The e optionalargument
paramscontainsalistofparametersofdist. Forexample,foraquantileplotofthe
uniformdistributionon[2,4]andx,use
qqplot (x, "unif", 2, 4)
distcanbeanystringforwhichafunctiondistinv ordist
inv existsthatcalculates
theinverseCDFofdistributiondist.
Ifnooutputargumentsaregiven,thedataareplotteddirectly.
[FunctionFile]
[p, y] ] = = ppplot
(
x
,
dist
,
params
)
PerformaPP-plot(probabilityplot).
IfFistheCDFofthedistributiondistwithparametersparamsandxasamplevector
oflengthn,thePP-plotgraphsordinatey(i)=F(i-thlargestelementofx)versus
abscissap(i)=(i-0.5)/n. IfthesamplecomesfromF,thepairswillapproximately
followastraightline.
Thedefaultfordististhestandardnormaldistribution.
Theoptionalargumentparams containsalistofparametersofdist.
Forexample,foraprobabilityplotoftheuniformdistributionon[2,4]andx,use
ppplot (x, "uniform", 2, , 4)
dist can n be any y string g for r which a a function n dist
cdf that t calculates the CDF of
distributiondistexists.
Ifnooutputisrequestedthenthedataareplottedimmediately.
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How to Save Acquired Image to File in C#.NET with in frames) { if (frame == TwainStaticFrameSizeType.LetterUS) { this.device.FrameSize = frame; break; } } }.
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610
GNUOctave
26.4 CorrelationandRegressionAnalysis
[FunctionFile]
cov
(
x
)
[FunctionFile]
cov
(
x
,
opt
)
[FunctionFile]
cov
(
x
,
y
)
[FunctionFile]
cov
(
x
,
y
,
opt
)
Computethecovariancematrix.
Ifeachrow of x andy y is s an n observation, andeachcolumnis s a variable, , thenthe
(i,j)-thentryofcov(x,y)isthecovariancebetweenthei-thvariableinxandthe
j-thvariableiny.
ij
=
1
N 1
XN
i=1
(x
i
¯x)(y
i
¯y)
where¯xand¯yarethemeanvaluesofxandy.
Ifcalledwithoneargument,computecov(x,x),thecovariancebetweenthecolumns
ofx.
Theargumentoptdeterminesthetypeofnormalizationtouse.Validvaluesare
0:
normalizewithN 1,providesthebestunbiasedestimatorofthecovari-
ance[default]
1:
normalizewithN,thisprovidesthesecondmomentaroundthemean
CompatibilityNote::Octavealwayscomputesthecovariancematrix. Fortwoinputs,
however,matlabwillcalculatecov(x(:),y(:))wheneverthenumberofelements
inx andy areequal. . Thiswillresultinascalarratherthanamatrixoutput. Code
relyingonthisodddefinitionwillneedtobechangedwhenrunninginOctave.
Seealso: [corr],page610.
[FunctionFile]
corr
(
x
)
[FunctionFile]
corr
(
x
,
y
)
Computematrixofcorrelationcoefficients.
If each row of x x and y is s an n observation and each columnis avariable, thenthe
(i,j)-thentryofcorr(x,y)isthecorrelationbetweenthei-thvariableinxandthe
j-thvariableiny.
corr(x;y)=
cov(x;y)
std(x)std(y)
If called with one argument, , compute e corr(x,x), , the correlation n between the
columnsofx.
Seealso: [cov],page610.
[FunctionFile]
spearman
(
x
)
[FunctionFile]
spearman
(
x
,
y
)
ComputeSpearman’srankcorrelationcoefficientrho.
For two data vectors x x andy, , Spearman’s s rho is s the e correlationcoefficient of the
ranksofx andy.
Chapter26: Statistics
611
Ifxandyaredrawnfromindependentdistributions,rhohaszeromeanandvariance
1/(n-1),andisasymptoticallynormallydistributed.
spearman(x)isequivalenttospearman(x,x).
Seealso: [ranks],page607,[kendall],page611.
[FunctionFile]
kendall
(
x
)
[FunctionFile]
kendall
(
x
,
y
)
ComputeKendall’stau.
Fortwodatavectorsx,yofcommonlengthn,Kendall’stauisthecorrelationofthe
signsofallrankdifferencesofxandy;i.e.,ifbothxandyhavedistinctentries,then
 =
1
n(n 1)
X
i;j
sign(q
i
q
j
)sign(r
i
r
j
)
inwhichtheq
i
andr
i
aretheranksofx andy,respectively.
Ifxandyaredrawnfromindependentdistributions,Kendall’stauisasymptotically
normalwithmean0andvariance
2(2n+5)
9n(n 1)
.
kendall(x)isequivalenttokendall(x,x).
Seealso: [ranks],page607,[spearman],page610.
[FunctionFile]
[theta, beta, dev, dl, , d2l, p] = = logistic_regression
(
y
,
x
,
print
,
theta
,
beta
)
Performordinallogisticregression.
Supposeytakesvaluesinkorderedcategories,andletgamma_i(x)bethecumulative
probabilitythaty fallsinoneofthefirsticategoriesgiventhecovariatex.Then
[theta, beta] ] = = logistic_regression n (y, , x)
fitsthemodel
logit (gamma_i i (x)) ) = = theta_i - beta’ * x,
i = = 1 ... k-1
Thenumberofordinalcategories,k,istakentobethenumberofdistinctvaluesof
round(y). Ifk k equals 2,y isbinaryandthemodelis ordinarylogisticregression.
Thematrixxisassumedtohavefullcolumnrank.
Giveny only,theta=logistic_regression(y)fitsthemodelwithbaselinelogit
oddsonly.
Thefullformis
[theta, beta, , dev, , dl, d2l, , gamma]
= logistic_regression n (y, , x, print, , theta, , beta)
inwhichalloutputargumentsandallinputargumentsexcepty areoptional.
Setting print to1requestssummary informationaboutthefittedmodeltobedis-
played. Settingprint t to2requestsinformationaboutconvergenceateachiteration.
Othervaluesrequestnoinformationtobedisplayed. Theinputargumentsthetaand
betagiveinitialestimatesforthetaandbeta.
Thereturnedvaluedev holdsminustwicethelog-likelihood.
The returned d values dl l and d d2l l are e the e vector of first and the matrix of second
derivativesofthelog-likelihoodwithrespecttothetaandbeta.
pholdsestimatesfortheconditionaldistributionofy givenx.
612
GNUOctave
26.5 Distributions
Octavehas functionsforcomputingtheProbabilityDensityFunction(PDF),theCumu-
lativeDistributionfunction(CDF),andthequantile(theinverseoftheCDF)foralarge
numberofdistributions.
Thefollowingtablesummarizesthesupporteddistributions(inalphabeticalorder).
Distribution
PDF
CDF
Quantile
Beta
betapdf
betacdf
betainv
Binomial
binopdf
binocdf
binoinv
Cauchy
cauchy
pdf
cauchy
cdf
cauchy
inv
Chi-Square
chi2pdf
chi2cdf
chi2inv
UnivariateDiscrete
discrete
pdf
discrete
cdf
discrete
inv
Empirical
empirical
pdf
empirical
cdf
empirical
inv
Exponential
exppdf
expcdf
expinv
F
fpdf
fcdf
finv
Gamma
gampdf
gamcdf
gaminv
Geometric
geopdf
geocdf
geoinv
Hypergeometric
hygepdf
hygecdf
hygeinv
KolmogorovSmirnov
NotAvailable
kolmogorov
NotAvailable
smirnov
cdf
Laplace
laplace
pdf
laplace
cdf
laplace
inv
Logistic
logistic
pdf
logistic
cdf
logistic
inv
Log-Normal
lognpdf
logncdf
logninv
UnivariateNormal
normpdf
normcdf
norminv
Pascal
nbinpdf
nbincdf
nbininv
Poisson
poisspdf
poisscdf
poissinv
StandardNormal
stdnormal
pdf
stdnormal
cdf
stdnormal
inv
t(Student)
tpdf
tcdf
tinv
UniformDiscrete
unidpdf
unidcdf
unidinv
Uniform
unifpdf
unifcdf
unifinv
Weibull
wblpdf
wblcdf
wblinv
[FunctionFile]
betapdf
(
x
,
a
,
b
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
Betadistributionwithparametersaandb.
[FunctionFile]
betacdf
(
x
,
a
,
b
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
theBetadistributionwithparametersaandb.
[FunctionFile]
betainv
(
x
,
a
,
b
)
Foreachelementofx,computethequantile(theinverseoftheCDF)atxoftheBeta
distributionwithparametersaandb.
Chapter26: Statistics
613
[FunctionFile]
binopdf
(
x
,
n
,
p
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
binomialdistributionwithparametersnandp,wherenisthenumberoftrialsand
pistheprobabilityofsuccess.
[FunctionFile]
binocdf
(
x
,
n
,
p
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thebinomialdistributionwithparameters nandp,wherenisthenumberoftrials
andpistheprobabilityofsuccess.
[FunctionFile]
binoinv
(
x
,
n
,
p
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
binomialdistributionwithparametersnandp,wherenisthenumberoftrialsand
pistheprobabilityofsuccess.
[FunctionFile]
cauchy_pdf
(
x
)
[FunctionFile]
cauchy_pdf
(
x
,
location
,
scale
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
Cauchydistributionwithlocationparameterlocationandscaleparameterscale >0.
Defaultvaluesarelocation=0,scale=1.
[FunctionFile]
cauchy_cdf
(
x
)
[FunctionFile]
cauchy_cdf
(
x
,
location
,
scale
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
theCauchydistributionwithlocationparameterlocationandscaleparameterscale.
Defaultvaluesarelocation=0,scale=1.
[FunctionFile]
cauchy_inv
(
x
)
[FunctionFile]
cauchy_inv
(
x
,
location
,
scale
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
Cauchydistributionwithlocationparameterlocationandscaleparameterscale.
Defaultvaluesarelocation=0,scale=1.
[FunctionFile]
chi2pdf
(
x
,
n
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
chi-squaredistributionwithndegreesoffreedom.
[FunctionFile]
chi2cdf
(
x
,
n
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
thechi-squaredistributionwithndegreesoffreedom.
[FunctionFile]
chi2inv
(
x
,
n
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
chi-squaredistributionwithndegreesoffreedom.
[FunctionFile]
discrete_pdf
(
x
,
v
,
p
)
For each h element t of x, , compute the probability density y function n (PDF) ) at x x of f a
univariatediscretedistributionwhichassumesthevaluesinv withprobabilitiesp.
614
GNUOctave
[FunctionFile]
discrete_cdf
(
x
,
v
,
p
)
Foreachelementofx,computethecumulativedistributionfunction(CDF)atx ofa
univariatediscretedistributionwhichassumesthevaluesinv withprobabilitiesp.
[FunctionFile]
discrete_inv
(
x
,
v
,
p
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
univariatedistributionwhichassumesthevaluesinv withprobabilitiesp.
[FunctionFile]
empirical_pdf
(
x
,
data
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
empiricaldistributionobtainedfromtheunivariatesampledata.
[FunctionFile]
empirical_cdf
(
x
,
data
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
theempiricaldistributionobtainedfromtheunivariatesampledata.
[FunctionFile]
empirical_inv
(
x
,
data
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
empiricaldistributionobtainedfromtheunivariatesampledata.
[FunctionFile]
exppdf
(
x
,
lambda
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)at x ofthe
exponentialdistributionwithmeanlambda.
[FunctionFile]
expcdf
(
x
,
lambda
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
theexponentialdistributionwithmeanlambda.
Theargumentscanbeofcommonsizeorscalars.
[FunctionFile]
expinv
(
x
,
lambda
)
For eachelement of x,computethe quantile (the inverseofthe CDF) at x x of f the
exponentialdistributionwithmeanlambda.
[FunctionFile]
fpdf
(
x
,
m
,
n
)
Foreachelementofx,computetheprobabilitydensityfunction(PDF)atxoftheF
distributionwithmandndegreesoffreedom.
[FunctionFile]
fcdf
(
x
,
m
,
n
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
theFdistributionwithmandndegreesoffreedom.
[FunctionFile]
finv
(
x
,
m
,
n
)
Foreachelementofx,computethequantile(theinverseoftheCDF)atx oftheF
distributionwithmandndegreesoffreedom.
[FunctionFile]
gampdf
(
x
,
a
,
b
)
For each h element t of x, , return the probability y density function n (PDF) ) at t x of f the
Gammadistributionwithshapeparameteraandscaleb.
[FunctionFile]
gamcdf
(
x
,
a
,
b
)
Foreachelement ofx,computethecumulativedistributionfunction(CDF)atx of
theGammadistributionwithshapeparameteraandscaleb.
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