MVR4 I&Range:6#'%"LG%%listG6"6#%@July~5,~1993~--~Zaven~A.~KarianGF(,&-%$MaxG6#9$"""- %$MinGF.!""6#%4Range~of~the~list~LGF(F(, I)SignTest:6#'%"CG%(booleanG6%%"iG%"kG%%varsG6#%VJoshua~D.~Levy~~|ir~~June~17,~ 1994,~modified~6/1/96-zakGF(C'@$4-%%typeG6$7#&9"6#;""#9#-F'6#/%%nameGF'-%&ERROR G6#%Yall~arguments~beyond~first~must~be~of~the~form~name=listG>8&-%'selectG6%FF -%'indetsGF.FR@$/FX<"-FT6#%Etest~expression~contains~no~variableG@$0FX-Fgn6#FH- FT6#%Ytest~expression~variable(s)~must~be~defined~in~argumentsG-%%subsG6%/%%tru eGF0/%&falseG""!-%(convertG6$7#-%$seqG6$-%&evalbG6#-%&evalfG6#-Ffo6$-Fbp6$/-%$l hsG6#&FJ6#8$&-%$rhsGFaq6#8%/FdqFLF//Fiq;F0-%%nopsG6#-Fgq6#&FJ6#FM%"+G6#%RNumber ~of~elements~in~list(s)~for~which~C~is~trueGF(F(, I-ScatPlotLine:F(63%"XG%"YG%"NG%"AG%%MinxG%%MaxxG%%MinyG%%MaxyG%#XRG%#YRG%#EXG% #EYGF=%"xG%#p1G%#p2GF&6#%CAugust~21,~1993~--~Zaven~A.~KarianGF(C5@$3/FNF04-FF6$ &FJ6#F0%)listlistG-FT6#%VWhen~one~argument~is~used,~it~must~be~a~list~of~listsG @$3/FNFM43-FF6$FbtF'-FF6$FbrF'-FT6#%PWhen~two~arguments~are~used,~both~must~be~ listsG@$3Fjt0-F^r6#Fbt-F^rFar-FT6#%HThe~two~lists~must~have~the~same~lengthG@%F ^tC$>Fdq7#-Fbp6$&&Fbt6#80Fct/Ffv;F0Fgu>Fiq7#-Fbp6$&FdvFcrFgvC$>FdqFbt>FiqFbr>FX -F^rFcq>8'-%&arrayG6#;F0,$FXFM?(FfvF0F0FXFioC$>&Fdw6#,&FfvFMF3F0&FdqFev>&Fdw6#, $FfvFM&FiqFev>8(-F2Fcq>8)-F-Fcq>8*-F2Fhq>8+-F-Fhq>8.,&Fjx#F0"#?Fgx#F3Ffy>8/,&F` yFeyF]yFgy>8,;,&FgxF0FcyF3,&FjxF0FcyF0>8-;,&F]yF0FiyF3,&F`yF0FiyF0>84-%'LinRegG 6%FdqFiq81>82-%%plotG6&7#-Fbp6$7$F]xFbx/Ffv;F0FXF\zFaz/%&styleG%&POINTG>83-F^[l 6%Ffz/FjzF\zFaz-%(displayG6#<$F\[lFj[l6#%UProduce~scatter~plot~of~X-Y~pairs~wit h~the~reg.~lineGF(F(, I'Anova1:6#'%"DGF'60%"mG%"nG%$totG%$SSEG%$SSTG%%SSTOGF\s%#PVG%%XBarGF=%"jG%*Gra ndMeanG%$MSTG%$MSEG6#%@July~7,~1993~--~Zaven~A.~KarianGF(C=>Fdq-F^rF.>FiqF\p>FX F\p>FdwF\p>FgxF\p>FgxF\p>F]y-Ffw6#;F0Fdq>F\zFa^l?(FazF0F0FdqFioC%>&F]y6#Faz-F^r 6#&F/Fi^l>&F\zFi^l-%%MeanGF[_l>Fiq,&FiqF0Fh^lF0?(FazF0F0FdqFio?(FcyF0F0Fh^lFioC $>FX,&FXF0&F\_l6#FcyF0>Fdw,&FdwF0*$,&Fh_lF0F^_lF3FMF0>Fiy*&FXF0FiqF3?(FazF0F0Fd qFio>Fgx,&FgxF0*&Fh^lF0,&F^_lF0FiyF3FMF0>Fjx,&FdwF0FgxF0>Ffv*&FgxF0,&FdqF0F3F0F 3>Fjz*&FdwF0,&FiqF0FdqF3F3>F`y,&F0F0-%%FCDFG6%Fi`lF\al*&FfvF0FjzF3F3-%'lprintG6 #%2~~~~~~~~~~~Sum~ofG-Fdal6#%I~~~~~~~~~~~Squares~~~Degrees~of~~Mean~SqG-Fdal6#% hnSource~~~~~~(SS)~~~~~~Freedom~~~~~(MS)~~~~~~F-Ratio~~~p-valueG-Fdal6#%hn----- --------------------------------------------------------G-%'printfG6(%NTreat.~~ %10.4f~~~%5d~~~%10.4f~~~%8.4f~~~%8.4fGFgxFi`lFfvFbalF`y-FdalF(-Fabl6&%>Error~~~ %10.4f~~~%5d~~~%10.4fGFdwF\alFjzFdbl-Fabl6%%5Total~~~%10.4f~~~%5dGFjx,&FiqF0F3F 0FdblF]bl6#%1One~factor~ANOVAGF(F(, I*DiscreteS:F(60F\sF]s%"BG%#llG%#ulG%$ptrG%$varG%"FGF=%"rGFb]lF&%%exprGF[]lF)F( C/@$3Fjt4F]u-FT6#%VWhen~two~arguments~are~used,~the~first~must~be~a~listG@$3Fjt 4-FF6$Fbr%'posintG-FT6#%]oWhen~two~arguments~are~used,~the~second~must~be~a~pos itive~integerG@$3/FN""$4-FF6$Fbt%*algebraicG-FT6#%inWhen~three~arguments~are~us ed,~the~first~must~be~an~expressionG@$3F[el4-FF6$Fbr%&rangeG-FT6#%ZWhen~three~a rguments~are~used,~the~second~must~be~a~rangeG@$3F[el4-FF6$&FJ6#F\elFedl-FT6#%^ oWhen~three~arguments~are~used,~the~third~must~be~a~positive~integerG@%F[elC+>F fvFbt>FjzFbfl>F]y-%#opG6#-FZ6%FF-FgnFevFR>Fdw-F`qFar>FgxF`r>FX-Ffw6#;F0,(FgxFMF dw!"#FMF0>FjxF0?(F\zFdwF0FgxFioC%>&FX6#FjxF\z>&FX6#,&FjxF0F0F0-Ffo6$/F]yF\zFfv> Fjx,&FjxF0FMF0>Fiy-F^p6$FXF'C$>FiyFbt>FjzFbr>Fdq,$-F^r6#Fiy#F0FM>Fiq-Ffw6#;F0Fj z>F`yFa^l>&F`yFct&FiyFcr?(F\zFMF0FdqFio>&F`y6#F\z,&&F`y6#,&F\zF0F3F0F0&Fiy6#,$F \zFMF0?(F\zF0F0FjzFioC&>Faz-%$rngGF(@$2FazF[jl>&FiqF`jl&FiyFct@$2&F`yFcqFaz>F`[ m&Fiy6#,&FdqFMF3F0?(FcyFMF0FdqFio@$32Faz&F`yFi_l1&F`y6#,&FcyF0F3F0Faz>F`[m&Fiy6 #,&FcyFMF3F07#-Fbp6$F`[m/F\zFhil6#%Gn~rand.~obs.~from~dist.~described~by~LGF(F(, I+ChisquareP:6$'FgclF`el'%"pGF`el6&%$eqnG%"aGFfs%"yG6#%CAugust~12,~1993~--~Zave n~A.~KarianGF(C'@$43-FF6$F/%(numericG-FF6$9%F^^m-%'RETURNG6#.-9!6#FJ@$52Fa^mF\p 2F0Fa^m-FT6#%HSecond~argument~must~be~between~0~and~1G>Fiq*&-%&GAMMAG6#,$F/Fdil F3)FMFe_mF3>Fdq/Fa^m-%$intG6$*(FiqF0)FX,&F/FdilF3F0F0-%$expG6#,$FX#F3FMF0/FX;F\ pFdw-%'fsolveG6%FdqFdw/Fdw;F\p%)infinityG6#%6Chi~square~percentileGF(F(, I-ChisquarePDF:6$F_]m'FfsF`el6#Fd]m6#%CAugust~11,~1993~--~Zaven~A.~KarianGF(C&@ $3F\^m1F/F\p-FT6#%NFirst~argument~must~be~a~variable~or~positiveG@$3F_^m1Fa^mF\ p-Fc^m6#F\p>FdqFa_m*(FdqF0)Fa^mF^`mF0-F``m6#,$Fa^mFc`mF06#%3Chi~square~densityG F(F(, I#FP:6%'%$df1GF`el'%$df2GF`elF`]m6'Fc]m%$numG%$denGFfsFe]mFf]mF(C)@$43F[^m-FF6$ 9&F^^mFb^m@$52FicmF\p2F0Ficm-FT6#%GThird~argument~must~be~between~0~and~1G@$52F /F\pF[_m-FT6#%EFirst~two~arguments~must~be~positiveG>Fiq*()FdwF^`mF0-Fc_m6#,&F/ FdilFa^mFdilF0)*&F/F0Fa^mF3Fe_mF0>FX*(Fb_mF0-Fc_m6#,$Fa^mFdilF0),&F0F0*(F/F0Fdw F0Fa^mF3F0F\emF0>Fdq/Ficm-Fj_m6$*&FiqF0FXF3/Fdw;F\pFgx-Fg`m6%FdqFgx/FgxFj`m6#%- F~percentileGF(F(, I+ChisquareS:6$'FgclFedl'F[]lFedl6'F=Fb]lFjrF]s%"tG6#%@May~27,~1994~--~Zaven~A. ~KarianGF(C'>Fdw-Ffw6#;F0Fa^m>FX-%%N01SG6#*&F/F0Fa^mF0>FiqF0?(FdqF0F0Fa^mFioC$> &FdwFcq-F^p6$7#-Fbp6$*$&FX6#FgxFM/Fgx;Fiq,(FiqF0F/F0F3F0Fdr>Fiq,&FiqF0F/F07#-Fb p6$Fjgm/FdqF`gm6#%FGenerates~sample~from~standard~normalGF(F(, I-ChisquareCDF:6$F_]m'Fe]mF`el6$Fd]mFfs6#%BAugust~8,~1993~--~Zaven~A.~KarianGF( @%3F_^mF[_mF\pC$>FdqFa_m-Fhp6#-Fj_m6$*(FdqF0)FiqF^`mF0-F``m6#,$FiqFc`mF0/Fiq;F\ pFa^m6#%/Chi~square~CDFGF(F(, I#FS:6%'%#n1GFedl'%#n2GFedlFgfm6%F=F]sFaclFf]mF(C%>Fiq-Fcfm6$F/Ficm>FX-Fcfm6$Fa ^mFicm7#-Fbp6$**&FiqFcqF0Fa^mF0F/F3&FXFcqF3/Fdq;F0Ficm6#%8Generates~sample~from ~FGF(F(, I*randomize:F(F(F)F(>%&_seedG-%&roundG6#,$-%%timeGF("**ycM76#%FdqFbt>FiyFbr@$F[el>Faz- %%sqrtG6#Fbfl>FjxFbw>F]y-F^r6#&FdqFct>Fcy,&FdilF0Fiy#F0"$+#@%5F[el2"#CF]y>Fiq-% (NormalPG6%F\pF0Fcy>Fiq-%#TPG6$,&F]yF0F3F0Fcy>Fdw-Ffw6#;F0,$FjxFM>FfvF0?(FgxF0F 0FjxFioC(>FX&FdqFbhm>F`y-F`_l6#FX@%Fjt>F\z-%&StDevGFdan>F\zFaz>&FdwFev-Fhp6#,&F `yF0*(FiqF0F\zF0-F`_n6#F]yF3F3>&Fdw6#,&FfvF0F0F0-Fhp6#,&F`yF0F_bnF0>Ffv,&FfvF0F MF07#-Fbp6$7$&Fdw6#,&FgxFMF3F0&Fdw6#,$FgxFM/Fgx;F0Fjx6#%RCompute~end-points~of~ confidence~intervals~for~muGF(F(, I+RunningSum:F$6%F\sF]sF=6#%CAugust~23,~1993~--~Zaven~A.~KarianGF(C'>FdqFj]l>Fi qFa^l>&FiqFct&F/Fct?(FXFMF0FdqFio>&FiqFdan,&&Fiq6#,&FXF0F3F0F0&F/FdanF07#-Fbp6$ Ffdn/FXFc^l6#%6Running~sum~of~a~listGF(F(, I0NormalVarianceS:6&'Fd]mF^^m'%"bGF^^mFgfm'Fj\lFedl6%FjrF[sF=Ff]mF(C&@$F^bm-FT6 #%ASecond~argument~must~be~positiveG>Fiq-Ffw6#;F09'?(FXF0F0FcfnFioC$>Fdq-%(Norm alSG6%F/Fa^mFicm>Ffdn-%)VarianceGFcq7#-Fbp6$Ffdn/FXFbfn6#%@Generates~variances~ from~normalGF(F(, I.NormTransVarS:FdenFienFf]mF(C&@$F^bmF\fn>FiqF`fn?(FXF0F0FcfnFioC$>FdqFgfn>Ffd n*(,&FicmF0F3F0F0F[gnF0Fa^mF3F]gn6#%LGenerates~transformed~variances~from~norma lGF(F(, I/ExponentialPDF:6$'FifmF`elFaamF(FcamF(C%@$FgamFiam@$FfimF_bm*&F/F3-F``m6#,$*& Fa^mF0F/F3F3F06#%5Exponentiall~densityGF(F(, I/ExponentialCDF:6$FchnFaim6#FfsFcimF(@%FfimF\p-Fhp6#-Fj_m6$*&F/F3-F``m6#,$*&Fd qF0F/F3F3F0/FdqFcjm6#%0Exponential~CDFGF(F(, I,BinomialPDF:6%'F\sF`elF`]mFaamF(Ff]lF(C(@$3F\^m4-FF6$F/%(integerG-FT6#%XFirst ~argument~must~be~a~variable~or~a~positive~integerG@$Fgam-FT6#%YFirst~argument~ must~be~a~variable~or~a~~positive~integerG@$3F_^mFj^m-FT6#%YSecond~argument~mus t~be~an~expression~or~between~0~and~1G@$33FgcmF\^m5F\dm2F/FicmF_bm@$3Fgcm54-FF6 $FicmFijnF\dmF_bm*(-%)binomialGFa[nF0)Fa^mFicmF0),&F0F0Fa^mF3,&F/F0FicmF3F06#%1 Binomial~densityGF(F(, I,Contingency:6#'F&-F'6#-F'6#%*nonnegintG6+%"hGF>F[]l%$nprG%$npcG%"qG%"cGF=Fb]l 6#%AJoshua~D.~Levy~~|ir~~June~21,~1994GF(C2>FiqFj]l>Fdq-F^r6#Fcdn>FX-%$sumG6$.- F^p6$&F/6#F`yFdr/F`y;F0Fiq>Fdw7#-Fbp6$-Fb^o6$&Fg^oF`jl/F\zFc^lFi^o>Fgx7#-Fbp6$- Fb^o6$&&F/6#.F`yF`jl/F\`oFj^oFb_o>FjxF\pFdbl?(F`yF0F0FiqFioC&?(F\zF0F0FdqFio-Fa bl6$%'%7.0f~GFa_o-Fabl6$%*~~%7.0f|+~G&FdwFh^o?(F\zF0F0FdqFioC%>F]y*(Fh`oF0&FgxF `jlF0FXF3>Fjx,&FjxF0-Fhp6#*&,&Fa_oF0F]yF3FMF]yF3F0-Fabl6$%((%6.2f)GF]yFdblFdbl? (F\zF0F0FdqFio-Fabl6$Fd`oF]ao-Fabl6$%'~%7.0fGFXFdblFdbl-Fabl6%%VChi-square~stat sistic~(%.0f~degrees~of~freedom):~%5g|+G*&Fi`lF0F[clF0Fjx-Fabl6$%/p-value:~%.6g |+G,&F0F0-F^im6$F`boFjxF3Fdbl6#%2Contingency~tableGF(F(, I,BinomialCDF:6%FbjnF`]mFaim6#F=FcimF(@'3FgcmF\dmF\p3Fh[oFj[oF0-Fhp6#-Fb^o6$*(- Fc\o6$F/FdqF0)Fa^mFdqF0)Ff\o,&F/F0FdqF3F0/Fdq;F\pFicm6#%-Binomial~CDFGF(F(, I-DiscUniformS:6$'%"RGFielFgfm6%F=Fd]mFgen6#%@July~6,~1993~--~Zaven~A.~KarianGF (C%>Fiq-F`qF.>FX-FgqF.7#-Fbp6$,&FiqF0-%&truncG6#*&,(FXF0FiqF3F0F0F0F[[mF0F0F[im 6#%GGenerates~sample~from~discrete~uniformGF(F(, I+ConfIntSuc:6$'F&Fdt'%&ValueG%)realconsG6'F[]l%&countGF=%$valGF`]n6#%AJune~13, ~1994~--~Zaven~A.~KarianGF(C(>Fdw-Fhp6#Fa^m>Fgx-FhpF.>Fdq-F^rFbhm>FiqF\p?(FXF0F 0FdqFio@$31Fdw&&FgxFdanFcr1&F`goFctFdw>Fiq,&FiqF0F0F0Fiq6#%Rnumber~of~confidenc e~intervals~that~contain~valueGF(F(, I-BernoulliPDF:6$F`]mFaamF(6#%BAugust~9,~1993~--~Zaven~A.~KarianGF(C%@$3F\^m5Fh am1F0F/-FT6#%UFirst~argument~must~be~a~variable~or~between~0~and~1G@$3F_^m5Fj^m 4-FF6$Fa^mFijnF_bm*&)F/Fa^mF0),&F0F0F/F3Ff\oF06#%2Bernoulli~densityGF(F(, I*BinomialS:6%'F\sFedl'Fa]mF^^mFgfm6%F]sF=F&Ff]mF(C'@$Fj^mF]_m>Fdq-Ffw6#;F0,&F/ FMFMF0?(FiqF\pF0F/FioC$>&Fdq6#,&FiqFMF0F0Fiq>&Fdq6#,&FiqFMFMF0*(-Fc\o6$F/FiqF0) Fa^mFiqF0)Ff\o,&F/F0FiqF3F0>FX7#-Fbp6$&FdqFhq/FiqF[jo-F^cl6$FXFicm6#%:n~random~ nos.~from~B(n,p)GF(F(, I-BernoulliCDF:6$F`]mFaimF\coFcimF(@'FfimF\p3F_^mF\_mF0-Fhp6#-Fb^o6$*&)F/FdqF0) F]io,&F0F0FdqF3F0F\jn6#%.Bernoulli~CDFGF(F(, I)GammaPDF:6%'Fd]mF`elFchnFaam6#FgenFcamF(C'@$FgamFiam@$F]bm-FT6#%OSecond~argum ent~must~be~a~variable~or~positiveG@$F^coF_bm>Fdq*&-Fc_mF.F3)Fa^mF/F3*(FdqF0)Fi cm,&F/F0F3F0F0-F``m6#,$*&FicmF0Fa^mF3F3F06#%.Gamma~densityGF(F(, I)GammaCDF:6%Fi\pFchnFaim6%FgenFfsF]sFcimF(C$@%F^co>FXF\pC$>FdqFc]p>FX-Fhp6#-Fj _m6$*(FdqF0)FiqFh]pF0-F``m6#,$*&FiqF0Fa^mF3F3F0/FiqF[doFX6#%*Gamma~CDFGF(F(, I,ContinuousS:6%'FhclF`elFadoFgfm62F=FeclFfsFfcl%$CDFGFifmFbclFccl%$ptsGF[sFb]l %%RandGFd]mFgenFh]o%"dG6#%DJanuary~24,~1994~--~Zaven~A.~KarianGF(C2>Fiq-F]gl6#F Y>F]yFcgl>F`yF`r>Fdw-Fj_m6$F//Fiq;F]yFjx>Fgx-&%(studentG6#%)makeprocG6$FdwFjx>F \z-Ffw6#;F0"#s>Faz-Ffw6#F\\n>FcyF\el?(FdqFMF0"#NFioC&>FX,&F]yF0*&Fi`lF0,&F`yF0F ]yF3F0#F0F]bp>&F\zFi_l-FgxFdan>&F\z6#,&FcyF0F0F0FX>Fcy,&FcyF0FMF0>&F\zFctF\p>&F \zFcrF]y>&F\z6#"#rF0>&F\z6#FgapF`y>Fiy-%$RNGG6#Ficm?(FdqF0F0FicmFioC)>FcyF0?(F( F0F0F(2&F\zFd\m&FiyFcq>FcyFjbp>Ffv&F\z6#,&FcyFM!"$F0>Fjz&F\z6#,&FcyFMFjglF0>F\[ lFadp>Fj[l&F\z6#,$FcyFM>&FazFcq,&FjzF0*(,&FbdpF0FfvF3F0,&F\[lF0FfvF3F3,&Fj[lF0F jzF3F0F07#-Fbp6$FcepF[\n6#%gnGen~sample~(size~n)~with~pdf~given~by~expr~on~the~ interval~RGF(F(, I,ConfIntProp:6$'F`]nFdt'Fe]nF\fo6-F&F]sF=Fb]lF[]lFj\lFd]nFa]nFa]m%"fGFdclFf]nF (C+@$5F[_m2Fa^nFa^mFb^n>FgxFj]l>FjxF^^o>F]y,&FdilF0Fa^mFi_n>F`y-Fa`n6%F\pF0F]y> Fiq-Ffw6#;F0Fdcn>FcyF0?(FXF0F0FgxFioC(>FdqF[en>F\z*&-Fb^o6$&Fdq6#Fdw/FdwFfcnF0F jxF3>Faz*&F`yF0-F`_n6#*(F\zF0,&F0F0F\zF3F0FjxF3F0>&FiqFi_l-Fhp6#,&F\zF0FazF3>&F iqFibp-Fhp6#,&F\zF0FazF0>FcyF\cp7#-Fbp6$7$&Fiq6#,&FXFMF3F0&Fiq6#Fiw/FX;F0Fgx6#% CCompute~confidence~intervals~for~pGF(F(, I)PoissonS:6$'%"lGF^^mFgfm6(F]s%(SumProbGF=%%ProbG%+UpperLimitGF&F)F(C,@$Fham-F T6#%@First~argument~must~be~positiveG>Fdq-Ffw6#;F0Fa^n>FiqF\p>FXF\p?(F(F0F0F(2F iq$"$***FhdpC'>Fdw-Fhp6#*()F/FXF0-F``m6#,$F/F3F0-%*factorialGFdanF3>Fiq,&FiqF0F dwF0>&Fdq6#,&FXFMF0F0FX>&Fdq6#,&FXFMFMF0Fdw>FX,&FXF0F0F0>Fc\qFX>Fg\q,&F0F0FiqF3 >FgxFi\q>Fjx7#-Fbp6$&FdqFdanFiip-F^cl6$FjxFa^m6#%An~random~nos.~from~Poisson~di st.GF(F(, I*FastCraps:6#Fgfm6(F]sFh\lF=%&pointG%%rollG%%gameGF\dnF(C%>Fdq-Ffw6#;F0F/?(FXF 0F0F/FioC'>Fiq-%$DieG6$""'FM>Fdw,&FbdnF0&FiqFcrF0@'5/Fdw""(/Fdw"#6>FjxF055/FdwF M/FdwF\el/Fdw"#7>FjxF3>FjxF\p?(F(F0F0F(/FjxF\pC%>FiqFh^q>FgxF]_q@&/FgxFdw>FjxF0 /FgxFb_q>FjxF3@%/FjxF3>Fd]qF\p>Fd]qF07#-Fbp6$Fd]q/FXFd^q6#%Fiq*&FdwF0-F`_nFcrF3>Fdq/Ficm-Fhp6#,&$""&F3F0-%$erfGFhqFgbq>FX-Fg`m6 $FdqFdw,&*&-F`_nFefoF0FXF0F0F/F06#%2Normal~percentileGF(F(, I(NormalS:6%'FhaqF^^m'FeclF^^mFgfm6+Fb]lF=%#P2G%"uG%"vGFd]mFgenFjr%)sqrt_varGFf ]mF(C)@$F^bmF\fn>F`y-Ffw6#;F0,&FicmF0F0F0>FdqF0>FX-Fhp6#,$%#PiGFM>F\zF`cq?(FiqF 0F0,&FicmFdilFdilF0FioC*>FdwF[[m>FgxF[[m>Fjx*&-F`_n6#,$-%$logGF]hpFjglF0-%$sinG 6#*&FXF0FgxF0F0>Fd[m-Fhp6#,&*&F\zF0FjxF0F0F/F0>Fdq,&FdqF0F0F0>F]y*&FbeqF0-%$cos GFieqF0>Fd[m-Fhp6#,&*&F\zF0F]yF0F0F/F0>FdqFafq7#-Fbp6$&F`yFhq/FiqF\\n6#%=Genera tes~sample~from~normalGF(F(, I)TimePlot:F(6*F\sFaclF[]lF=Fb]lFifmFc]nF]s6#%CAugust~19,~1993~--~Zaven~A.~Kari anGF(C)>FdqFN>FXFgu>F]yFjn?(FdwF0F0FdqFioC$@$4-FF6$&FJF]hpF'-FT6#%FXFhhq>FiqFew?(FdwF0F0FdqFioC%>F`yFbhq?(FgxF0F0 -F^rFh^oFioC$>&FiqF`cnFgx>&FiqFccn&F`yFbhm>F]y-%&unionG6$F]y<#7#-Fbp6$7$&Fiq6#, &FjxFMF3F0&Fiq6#F\an/Fjx;F0F`iq-F^[lFabn6#%;Time~series~plot~of~a~listGF(F(, I/UniformMedianS:6%FadoFgfmFhenFienFf]mF(C&@$543-FF6$FhdoF^^m-FF6$FjdoF^^m2Fjdo Fhdo-FT6#%GFirst~argument~must~be~a~numeric~rangeG>FiqFiap?(FXF0F0FicmFioC$>Fdq -%)UniformSG6$F/Fa^m>Ffdn-%+PercentileG6$FdqFgbq7#-Fbp6$Ffdn/FXF\\n6#%JGenerate s~medians~from~continuous~uniformGF(F(, I)UniformP:6$FadoF`]m6%Fd]mFgenFfsFf]mF(C(>FdqFhdo>FiqFjdo@$433-FF6$FdqF^^m-FF6 $FiqF^^mF_^mFb^m@$Fj^mF]_m@$2FiqFdq-FT6#%?First~argument~must~be~a~rangeG-Fg`m6 $/*&,&FXF0FdqF3F0F\alF3Fa^mFX6#%3uniform~percentileGF(F(, I)UniformS:F`do6%Fd]mFgenF=Ff]mF(C&>FdqFhdo>FiqFjdo@$54Fe]rF\^r-FT6#%NFirst~arg ument~must~be~a~proper~numeric~rangeG7#-Fbp6$-Fhp6#,&*&F\alF0F[[mF0F0FdqF0/FXF` gm6#%IGenerates~sample~from~continuous~uniformGF(F(, I&StDev:F$F(F)F(-F`_n6#-F\gnF.6#%AStandard~deviation~of~the~list~LGF(F(, I2HypergeometricPDF:6&'FjjmF`el'F\[nF`elF_]mFaamF(6#%DJanuary~27,~1994~--~Zaven ~A.~KarianGF(C&@$33-FF6$FcfnF^^mFgcm2FicmFcfn-FT6#%YFourth~argument~cannot~be~l arger~than~the~third~argumentG@$33F_arF\^m2F/Fcfn-FT6#%YFourth~argument~cannot~ be~larger~than~the~first~argumentG@$33F^arF_^m2Fa^m,&FicmF0FcfnF3-FT6#%Z2nd~arg ument~cannot~be~smaller~than~the~3rd-4th~argumentsG*(-Fc\o6$F/FcfnF0-Fc\o6$Fa^m F`brF0-Fc\o6$,&F/F0Fa^mF0FicmF36#%7Hypergeometric~densityGF(F(, I)Kurtosis:F$6(%#sfG%"MGF\s%"VGF=FjrFfgqF(C)@%F`t>Fjx7#-Fbp6$-%"$G6$&&F/FbhmFct &F^drFcr/Fgx;F0Fj]l>FjxF/>FdqF\p>Fiq-F`_lF`hl>FX-F^rF`hl>Fdw*(FjdnF0-F\gnF`hlF0 FXF3?(FgxF0F0FXFio>Fdq,&FdqF0*$,&&FjxFbhmF0FiqF3""%F0*(FdqF0FXF3FdwFjgl6#%DKurt osis~or~3rd~standardized~momentGF(F(, I,PlotPolyReg:F(62FjrF[sF]sF=F>F[]lFaclF:Fb]lFb]n%#BIGFc]mFfsF\`pFh]oFj\lFi`rF( C5@$3FjtF_t-FT6#%inWhen~two~arguments~are~used,~the~first~must~be~a~list~of~lis tsG@$3F[elF[u-FT6#%enWhen~three~arguments~are~used,~the~first~two~must~be~lists G@$3F[elFfuFju>%'DigitsG"#]@%FjtC%>Fdq7#-Fbp6$&&FbtF]hpFct/FdwFhv>Fiq7#-Fbp6$&F `grFcrFagr>Fj[lFbrC%>FdqFbt>FiqFbr>Fj[lFbfl>FjxFbw>F]y-Ffw6$;F0,&Fj[lF0F0F0F`hr >F`y-Ffw6$F`hr;F0F0>&F]y6$F0F0Fjx?(FdwF0F0FahrFio?(F\zFdwF0FahrFioC'@$/F\zF0>F\ z,&F\zF0F0F0>FazF\p>Faz-F^p6$7#-Fbp6$)Faan,(FdwF0F\zF0FjglF0FecnFdr>&F]y6$FdwF\ 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F]yF'>Faz-Ffw6#Fjip?(FiqF0F0FgxFio>&FazFhqF\p>F`yF0>FiqF0?(F(F0F0F(1F`yFdq@%31& FicmFhq&FXFh^o2Fads&Ficm6#FdgoC$>Fhcs,&FhcsF0F0F0>F`y,&F`yF0F0F0>FiqFdgo@%F[bs> FfvF0>FfvF\p>FcyFdcs>&FcyFct-Fhp6#-Ffo6$/Fjx,&&FicmFcrF0FfvF3Fa^m?(FiqFMF0,&Fgx F0F3F0Fio>&FcyFhq-Fhp6#,&-Ffo6$/Fjx,&FcdsF0FfvF3Fa^mF0-Ffo6$/Fjx,&F`dsF0FfvF3Fa ^mF3>&FcyFbhm-Fhp6#,&F0F0-Ffo6$/Fjx,&&FicmFbhmF0FfvF3Fa^mF3>F\zF\p?(FiqF0F0FgxF io>F\z,&F\zF0*(,&FhcsF0*&FdqF0F[fsF0F3FMFdqF3F[fsF3F0?(FiqF0F0FgxFio@$2FggsFaer C$-Fabl6%%UWarning:~Class~%3.0f~has~an~expected~frequency~of~%fGFiqFggsFdbl@%F[ bs>FfvFgbq>FfvF\p@%F[bs>FfzFgx>FfzFdw>Fjz<#7&7$F_gsF\p7$F_gs*(&FazFbhmF0FdqF3,& FibsF0F`gsF3F37$,&&Ficm6#FfzF0FfvF0Fjhs7$F^isF\p?(FiqF0F0FiesFio>Fjz-Fiiq6$Fjz< #7&7$FffsF\p7$Fffs*(FhcsF0FdqF3,&FcdsF0F`dsF3F37$FbfsFjis7$FbfsF\p>F\[l-F^[l6%F jz/%&colorG%%blueG/%*linestyleGF\el@%F[bsC%>Fjz<#7&Fhhs7$F_gs*&FhfsF0F\isF37$,& F`gsF0FfvF0F\[t7$F^[tF\p?(FiqF0F0FiesFio>Fjz-Fiiq6$Fjz<#7&Fhis7$Fffs*&F[fsF0F[j 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ARNING:~There~areGF\[l%@~data~points~out~of~plot~range.G>F\zF0>Fdq-Ffw6#;F0Fdw> Fjx-Ffw6#;F0,&Fh_tF0F0F0>F]yFcat?(F`yF0F0FdwFio@$31Fj[lFc`t1Fc`tFfzC$>&FdqF`jlF c`t>F\zF_ir>FdwFdjl>FX7#-Fbp6$&FdqFi^l/FazFaat>Fcy-Ffw6#;F0,&Fh_tFMFaerF0>Fgx*& ,&FfzF0Fj[lF3F0Fh_tF3>&FjxFctFj[l>&F]yFctF\p?(F`yFMF0FfatFioC$>&FjxFh^o,&&Fjx6# ,&F`yF0F3F0F0FgxF0>&F]yFh^oF\p@$2&Fjx6#Ffat&FXF]hp>F`dtFbdt>F`yFM?(F\zF0F0FdwFi o@%1&FXF`jlFgct>&F]yFjct,&FjdtF0F0F0C$>F`yFids>F\zFdjl>&FfvFct,&Fj[lF0FgxFc`m>& FjzFctF\p>FiyFM?(F`yF0F0Fh_tFioC%>&FfvFcil,&FgctF0FgxFdil>&FjzFcilF]dt>Fiy,&Fiy F0F0F0>&Ffv6#,&Fh_tF0FMF0,&FfzF0FgxFdil>&FjzF`ftF\p?(F`yF0F0FaftFioC$>&Fcy6#,&F `yFMF3F0&FfvFh^o>&Fcy6#,$F`yFM*(&FjzFh^oF0FgxF3FdwF37#-Fbp6$7$FhftF]gt/F`y;F0Fa ft-F^[l6%Fe\tF\\tFdjs6#%=Graphic~display~of~histogramGF(F(, I+PlotEmpCDF:6$F%Fado6&F=F]sF\sFjrFcimF(C(>FXFj]l>Fdw-F_csF.>Fiq<#7$7$-%$minG6$ -F`qFefo&FdwFctF\p7$F\itF\p?(FdqF0F0FjdnFio@$2Fjgm&Fdw6#Fafq>Fiq-Fiiq6$Fiq<#7$7 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wF3-F^[l6$7#-Fbp6$7$F`bblFbbbl/FdwFc^lF\\t6#%GPlotting~the~running~average~of~a ~listGF(F(, I&BetaS:6%'FjjmF^^m'F\[nF^^mFgfm6'F=Fe]m%#yyGF]sFjrFf]mF(C(@$FhamFhjp@$F^bmF\fn >FgxFiap>Fdw-Ffjm6%,&Fa^mFMFMF0F\joFicm?(FdqF0F0FicmFioC%>FX*(Fjc_lF0FjgmF0Fgc_ lF3>Fiq*$F[]qF3>FbialFiq7#-Fbp6$FbialF[\n6#%;Generates~sample~from~betaGF(F(, I*ClassFreq:Fge`l6)F\sF=F]s%#SLG%%bdryG%&deltaGFdclFg^yF(C)@%Fjt>FdqFi_t>FdqFic m>FXFa^l>FdwFcht>Fjx*&,&FfjtF0F[itF3F0FdqF3>F]yF0?(FiqF0F0FdqFioC%>FjbsF\p>Fgx, &*&FiqF0FjxF0F0F[itF0?(F(F0F0F(31F]y-F^rF]hp1&FdwFabnFgxC$>Fjbs,&FjbsF0F0F0>F]y Ff`z7#-Fbp6$Fjbs/FiqFc^l6#%UClass~Frequencies~of~L~on~the~range~R~in~n~interval sGF(F(, I*NormalPDF:6%FgaqFiaqFaamFbamFcamF(C%@$F]bmF^]p>Fdq*$-F`_n6#,$*&FidqF0Fa^mF0FM F3*&FdqF0-F``m6#,$FdbvFc`mF06#%/Normal~densityGF(F(, I$GBP:6'F`czFbczFdczFfczF`]m6'Fc]mF:Ff]xFfsFe]mFf]mF(C*@$433FfcmF_arFh`vFb^m@$1 FicmF3-FT6#%=~1G@$1FcfnF3-FT6#%=Fourth~argument~must~b e~>=~1G@$52Fj`vF\pF]`al-FT6#%GFifth~argument~must~be~between~0~and~1G>FiqFedz>F XF_ez>Fdq/Fj`v-Fj_m6$**FiqF0FXF0)FgbalFicmF0),(F/F0Fa^mF0FdwF3FcfnF0/Fdw;F/Fgx- Fg`m6$FdqFgx6#%FdqFh`s@$5/-F^r6#7#FdqF\p2F0Fg[clFdas>FiqFc ht>FXFa_t>FdwF\p?(FgxF0F0FXFioC&>Fjx-Ffo6$/FdqF][sFa^m>F]y-Fhp6#-F_d\l6#,&*&Fie sF0FXF3F0FjxF3>F`y-Fhp6#-F_d\l6#,&*&FgxF0FXF3F0FjxF3@$2Fdw-Fdjt6$F]yF`yC$>FdwFd ]cl>F\zFgx>Faz-F\ht6$FiqFicm>Fcy-F^[l6%Fa^m/FdqFicmFajs-Fi\t6#-F_\l6#F\^al-Fabl 6$%KThe~K-S~goodness~of~fit~statistic~of~%7.4fGFdw-Fabl6$%2~occured~at~%6.4fGF` [mFdblFdbl6#%`uTo~perform~a~Kolmogorov-Smirnov~goodness~~~~~~~~~of~fit~test~for ~the~data~in~L,~the~distribution~~~~~~~~~with~CDF~expr,~on~the~range~of~values~ specified~~~~~~~~by~R.GF(F(, I*NormalCDF:6%FgaqFiaqFaim6#FifmFf]lF(C$>Fdq*&FebvF0-F`_n6#,$Fa^mFMF3-Fhp6#,&Fg bqF0-FjbqFcqFgbq6#%+Normal~CDFGF(F(, I*Wilcoxon2:6$F^dal'F[sFh[_l6*Fc]nF[]lFfsFe]mF\\_lF=Fb]lF>F@F(C'>Fdq-F_cs6$7$-F ]gl6#-Fidal6$FXFgfo-F]gl6#-Fidal6$FdwFdfo:F[]_lF(F\]_lF(F`falF(F(>FiqFbw>FgxF\p ?(FjxF0F0FiqFioC$?(F]yFjxF0F[cl/-F]glFf__l-F]gl6#&Fdq6#Ff`zF(@%/F]yFjx>Fgx,&Fgx F0*&-F^_^l6#-Ffo6%/FXF\p/FdwF0Fb^_lF0FjxF0F0C$>Fgx,&FgxF0*&,&FjxF0F]yF0F0-F^_^l 6#-Ffo6%FcbclFdbcl-Fb^o6$&FdqFh^o/F`y;FjxF]yF0Fdil>FjxF]yFgx6#%dqWilcoxon~test~ for~equality~of~two~medians.~Returns~sum~of~ranks~of~Y~in~the~sorted~combinatio n~of~X~and~YGF(F(, I$RNG:F[^qF\coF)F(7#-Fbp6$F`e`l/FdqFd^q6#%F@F(C(@'Fjt@%-FF6$,&Fj]lF0-F^rFefoF0%%evenG> Fgx,&Fj]lFdilFgdclFdil-FT6#%Hsum~of~numbers~of~observations~not~evenGF`fl>FgxFb fl-FT6#%.bad~argurmentG>Fdq-F_cs6$Ffeal:F[]_lF(F\]_lF(F`falF(F(>FiqF0>FXF\p>Fdq Fgfal-Fb^o6$F\hp/FdwFjip6#%,Median~testGF(F(, I,Convolution:6$'%#X1GF''%#X2GF'64%%dim1G%%dim2G%#l1G%#l2G%#j1G%#j2GF=%#L1G%#L2 G%#N1G%#N2GF]sF>F`ablFb]l%#f1G%#f2GFifm6#%ESeptember~7,~1993~--~Zaven~A.~Karian GF(C:>Fgx-Fb^o6$&F/6#,$F]yFM/F]y;F0,$Fj]lFdil>Fjx-Fb^o6$&Fa^mFigcl/F]y;F0,$Fgdc lFdil@$52$F0F[fy-F_d\lFb\al2Fhhcl-F_d\l6#,&FjxF0F3F0-FT6#%?Probabilities~must~a dd~up~to~1G?(F]yF0F0F]hclFio@$2&F/6#,&F]yFMF3F0F\p-FT6#%LRandom~variable~values ~must~be~non-negativeG?(F]yF0F0FdhclFio@$2&Fa^mFeiclF\pFgicl>Fdq,&Fj]lF0FcdnFM> Fiq,&FgdclF0&Fa^mFctFM>FXFa^l>FdwF_hz>FgxF0>FjxF0?(F]yF\pF0,&FcdnF0F3F0FioC%>Fa hmF\p>&FX6#F_galF\p>Fgx,&FgxF0FMF0?(F]yF0F0Fj]lFioC$>Fahm&F/Fabn>FgxF_gal?(F]yF \pF0,&FbjclF0F3F0FioC%>&FdwF`hlF\p>&FdwFchlF\p>FjxFihl?(F]yF0F0FgdclFioC$>Fi[dl &Fa^mFabn>FjxFdhl>F`y7#-Fbp6$Fchx/F]y;F0Fies>F\z7#-Fbp6$Fcfbl/F]y;F0F]icl>Faz,$ F`iqFdil>Fcy,$-F^rF`jlFdil>Fiy-Ffw6#;F0,(FazFMFcyFMFjglF0?(FfvF\pF0,(FazF0FcyF0 FjglF0FioC&>FjzF\p?(F\[lF\pF0FfvFioC%@%52F\[lF\p2,&FazF0F3F0F\[l>Fj[lF\p>Fj[l&F `y6#,&F\[lFMFMF0@%52,&FfvF0F\[lF3F\p2Fa\mF[_dl>FfzF\p>Ffz&F\z6#,(FfvFMF\[lFjglF MF0>Fjz,&FjzF0Ffe\lF0>&Fiy6#,&FfvFMF0F0Ffv>&Fiy6#,&FfvFMFMF0Fjz>F\[lF\p?(F(F0F0 F(/&Fiy6#,&F\[lF0FMF0F\p>F\[lFa`dl7#-Fbp6$&Fiy6#Fh_t/Fh_t;Ff`tFg]dl6#%YComputat ion~of~probabilities~via~the~convolution~formulaGF(F(, I-type/package:6#Fdfp6#Fifm6#Fc^]lFhi^lC$>F\_^lF]_^l@%3-FF6$F\_^l.Fd]]l-Fd^^l6$ 7#-F]gl6$""$F[`^lFa`^lFioF[pFhi^lFhi^lFhi^l, I(CauchyS:F[^qF\coFddoF(7#-Fbp6$-Fhp6#-%$tanG6#,&*&FidqF0F[[mF0F0FidqFc`mF[dcl6 #%=Generates~sample~from~CauchyGF(F(, I.NormalMedianS:FdenFienFf]mF(C&@$F^bmF\fn>FiqF`fn?(FXF0F0FcfnFioC$>FdqFgfn>Ffd nFb\rF]gn6#%>Generates~medians~from~normalGF(F(, I'GBDPDF:6'F`czFbczFdczFfczFaam6$F:Ff]xFcamF(C'@$3FgcmFihbl-FT6#%JThird~argumen t~must~be~a~variable~or~>~-1G@$3F_arF^ibl-FT6#%KFourth~argument~must~be~a~varia ble~or~>~-1G>FdqFedz>FiqF_ez**FdqF0FiqF0),&Fj`vF0F/F3FicmF0),(F/F0Fa^mF0Fj`vF3F cfnF06#%9Generalized~beta~densityGF(F(, I/GraphRandWalk2:6%FgfvFifvF[gv6(F]sF`gvF^gvF_gvFb]lF&F)F(C)>Fdq-Ffw6#;F0,$Ficm FM>FiqFdgv>FbdnF\p>F^_qF\p?(FgxF0F0FicmFioC(>FXF[[m>FdwF[[m@%F[\w>F^_qF]\w>F^_q F`\w@%1FdwFa^m>FbdnFd\w>FbdnFf\w>&FdqF`cnFbdn>&FdqFccnF^_q>Fjx7#-Fbp6$7$FhfdlFj fdl/FgxF\\n-F^[lF`hlFf]wF(F(, I/NegBinomialPDF:6%F_]mF`]mFaamF(Ff]lF(C'@$Fb[o-FT6#%VSecond~argument~must~be~a ~variable~or~between~0~and~1G@$Fejn-FT6#%PFirst~argument~must~be~a~variable~or~ an~integerG@$3FgcmF^\oF_bm@$3Fh[o2FicmF/F_bm*(-Fc\o6$F]hnFh]pF0Fe]pF0)Ff\oFebvF 06#%:Negative~binomial~densityGF(F(, I-NegBinomialS:6%FffmFdioFgfmFbjpFf]mF(C,@$Fj^mF]_m>FdqF\[q>FiqF\p>FXF/?(F(F0F0 F(Fb[qC'>Fdw-Fhp6#*(-Fc\o6$FjdnFh]pF0Fe]pF0)Ff\oFiezF0>FiqFa\q>&Fdq6#,(FXFMF/Fj glF0F0FX>&Fdq6#,(FXFMF/FjglFMF0Fdw>FXF[]q>F\jdlFX>F`jdlF^]q>FgxFbjdl>FjxFa]q-F^ cl6$FjxFicm6#%Kn~random~nos.~from~negative~binomial~dist.GF(F(, I/NegBinomialCDF:6%F_]mF`]mFaimF\coFcimF(@%FahdlF\p-Fhp6#-Fb^o6$*(-Fc\o6$Fi`lFh ]pF0Fe]pF0)Ff\o,&FdqF0F/F3F0/Fdq;F/Ficm6#%6Negative~binomial~CDFGF(F(, I+ConfIntVar:F`fp6-F&F]sF=F[]lFj\l%#ssGFdcl%%chi1G%%chi2G%%CHI1G%%CHI2GFf]nF(C- @$FgfpFb^n>FdwFj]l>FgxF^^o>F`y,&FdilF0Fa^m#F3Fj_n>F\z,&F0F0F`yF3>Faz*&FiesF0-F\ ]m6$FiesF`yF3>Fcy*&FiesF0-F\]mF`]tF3>Fiq-Ffw6#;F0F^]w>F]yF0?(FXF0F0FdwFioC'>Fdq F[en>FjxF[gn>Fh[]l*&FjxF0FcyF0>&FiqFiacl*&FjxF0FazF0>F]y,&F]yF0FMF07#-Fbp6$Fcip /FXFaat6#%JCompute~confidence~intervals~for~sigma**2GF(F(, I-GeometricPDF:FigoF(FcamF(C%@$3F\^mFd[\lFaho@$3F_^m5Fgho2Fa^mF0F_bm*&F/F0)F]io ,&Fa^mF0F3F0F06#%2Geometric~densityGF(F(, I-GeometricCDF:Fi[pF\coFcimF(@%3F_^mFc_elF\p-Fhp6#-Fb^o6$*&F/F0)F]ioFi`lF0F[im6 #%.Geometric~CDFGF(F(, I#TP:6$F^^vF`]m6(Fc]mFa^vFb^vFc^vFfsFe]mFf]mF(C)@$Fj]mFb^m@$Fj^mF]_m>FiqFg^v>FX F[_v>Fdw),&F0F0*&FgxFMF/F3F0Fi^v>Fdq/Fa^m-Fj_m6$*(FiqF0FXF3FdwF3/Fgx;FidvFjx-Fg `m6$FdqFjx6#%6Students~t~percentileGF(F(, I%Freq:F^ht6'F^foF\sF=F2F-Ff]mF(C*>FdwF[it>FgxFfjt@$543-FF6$FdwF^^m-FF6$FgxF^^m 2FgxFdw-FT6#%RSecond~argument~must~be~a~propoer~numerical~rangeG>FdqF]`x>FiqFj] l?(FXFdwF0FgxFio>Fd]qF\p?(FXF0F0FiqFio@$31FdwF[en1F[enFgx>&Fdq6#F[en,&FicelF0F0 F07#-Fbp6$Fd]q/FXF_`x6#%@Frequencies~of~L~on~the~range~RGF(F(, I#TS:6$'%#nuGFedlFgfmF][nFf]mF(C%>Fiq-FcgmFefo>FX-FcfmF`\r7#-Fbp6$*&Fi[nF0-F`_n 6#*&Fj[nF0F/F3F3F[im6#%BGenerates~sample~from~Student's~tGF(F(, I'Median:F$F(6#%@May~26,~1994~--~Zaven~A.~KarianGF(-Fc\r6$F/Fgbq6#%5The~1n~of~t he~list~LGF(F(, I-UniformMeanS:F\[rFienFf]mF(C&@$F_[rFg[r>FiqFiap?(FXF0F0FicmFioC$>FdqF^\r>Ffdn FfawFe\r6#%HGenerates~means~from~continuous~uniformGF(F(, I+GeometricS:F`[\lFbjpFf]mF(C)@$Fd[\lFf[\l>Fdq-Ffw6#;F0"$+&>FiqF\p>FXF0@%/F/F0C %>Ff_nF0>&FdqFcrF0>FgxFMC&?(F(F0F0F(Fb[qC'>Fdw*&F/F0)F]ioFjdnF0>FiqFa\q>&FdqFei pFX>&FdqFhipFdw>FXF[]q>FbhelFX>FdhelF^]q>FgxFiw>FjxFa]qFe]q6#%Cn~random~nos.~fr om~geometric~dist.GF(F(, I)RandWalk:6'FgfvF[[wFifvF[gvFgfm6'F]sF`gvF=Fb]lFgclF)F(C&>Fdq-Ffw6#;F0Fj`v>Fiq Fdgv?(FXF0F0Fj`vFioC&>FbdnF\p>F^_qF\p?(FdwF0F0FcfnFioC$>FgxF[[m@)1FgxF/>F^_qF]\ w1FgxF[cr>F^_qF`\w1FgxFb\w>FbdnFd\w>FbdnFf\w>Fd]q7$FbdnF^_q7#-Fbp6$Fd]q/FXFdiel FbivF(F(