Mathematical and statistical functions for the Empirical distribution, which is commonly used in sampling such as MCMC.
Returns an R6 object inheriting from class SDistribution.
The Empirical distribution is defined by the pmf, $$p(x) = \sum I(x = x_i) / k$$ for \(x_i \epsilon R, i = 1,...,k\).
Sampling from this distribution is performed with the sample function with the elements given as the support set and uniform probabilities. Sampling is performed with replacement, which is consistent with other distributions but non-standard for Empirical distributions. Use simulateEmpiricalDistribution to sample without replacement.
The cdf and quantile assumes that the elements are supplied in an indexed order (otherwise the results are meaningless).
The distribution is supported on \(x_1,...,x_k\).
Emp(samples = 1)
N/A
N/A
McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.
Other discrete distributions:
Arrdist,
Bernoulli,
Binomial,
Categorical,
Degenerate,
DiscreteUniform,
EmpiricalMV,
Geometric,
Hypergeometric,
Logarithmic,
Matdist,
Multinomial,
NegativeBinomial,
WeightedDiscrete
Other univariate distributions:
Arcsine,
Arrdist,
Bernoulli,
BetaNoncentral,
Beta,
Binomial,
Categorical,
Cauchy,
ChiSquaredNoncentral,
ChiSquared,
Degenerate,
DiscreteUniform,
Erlang,
Exponential,
FDistributionNoncentral,
FDistribution,
Frechet,
Gamma,
Geometric,
Gompertz,
Gumbel,
Hypergeometric,
InverseGamma,
Laplace,
Logarithmic,
Logistic,
Loglogistic,
Lognormal,
Matdist,
NegativeBinomial,
Normal,
Pareto,
Poisson,
Rayleigh,
ShiftedLoglogistic,
StudentTNoncentral,
StudentT,
Triangular,
Uniform,
Wald,
Weibull,
WeightedDiscrete
distr6::Distribution -> distr6::SDistribution -> Empirical
nameFull name of distribution.
short_nameShort name of distribution for printing.
descriptionBrief description of the distribution.
aliasAlias of the distribution.
Inherited methods
distr6::Distribution$cdf()distr6::Distribution$confidence()distr6::Distribution$correlation()distr6::Distribution$getParameterValue()distr6::Distribution$iqr()distr6::Distribution$liesInSupport()distr6::Distribution$liesInType()distr6::Distribution$median()distr6::Distribution$parameters()distr6::Distribution$pdf()distr6::Distribution$prec()distr6::Distribution$print()distr6::Distribution$quantile()distr6::Distribution$rand()distr6::Distribution$stdev()distr6::Distribution$strprint()distr6::Distribution$summary()distr6::Distribution$workingSupport()
new()Creates a new instance of this R6 class.
Empirical$new(samples = NULL, decorators = NULL)mean()The arithmetic mean of a (discrete) probability distribution X is the expectation $$E_X(X) = \sum p_X(x)*x$$ with an integration analogue for continuous distributions.
mode()The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).
variance()The variance of a distribution is defined by the formula $$var_X = E[X^2] - E[X]^2$$ where \(E_X\) is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.
skewness()The skewness of a distribution is defined by the third standardised moment, $$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution.
kurtosis()The kurtosis of a distribution is defined by the fourth standardised moment, $$k_X = E_X[\frac{x - \mu}{\sigma}^4]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.
entropy()The entropy of a (discrete) distribution is defined by $$- \sum (f_X)log(f_X)$$ where \(f_X\) is the pdf of distribution X, with an integration analogue for continuous distributions.
mgf()The moment generating function is defined by $$mgf_X(t) = E_X[exp(xt)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.
cf()The characteristic function is defined by $$cf_X(t) = E_X[exp(xti)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.
pgf()The probability generating function is defined by $$pgf_X(z) = E_X[exp(z^x)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.
setParameterValue()Sets the value(s) of the given parameter(s).
...ANY
Named arguments of parameters to set values for. See examples.
lst(list(1))
Alternative argument for passing parameters. List names should be parameter names and list values
are the new values to set.
error(character(1))
If "warn" then returns a warning on error, otherwise breaks if "stop".
resolveConflicts(logical(1))
If FALSE (default) throws error if conflicting parameterisations are provided, otherwise
automatically resolves them by removing all conflicting parameters.
## ------------------------------------------------
## Method `Empirical$new`
## ------------------------------------------------
Empirical$new(runif(1000))
#> Emp(data = list(samples = c(0.000103691359981894, 0.00164648867212236, 0.00247683003544807, 0.00249242829158902, 0.00401845388114452, 0.00408198102377355, 0.00449630804359913, 0.00720894057303667, 0.00750555354170501, 0.00865807477384806, 0.0108724641613662, 0.0132726018782705, 0.013695368077606, 0.0147188026458025, 0.0147690158337355, 0.0150322786066681, 0.0160031442064792, 0.0170156876556575, 0.0177699560299516, 0.0185314333066344, 0.0189061700366437, 0.019525250652805, 0.0196958889719099, 0.0221170505974442,
#> 0.0238767648115754, 0.0248860311694443, 0.0266246700193733, 0.0270194539334625, 0.028712565312162, 0.0291041533928365, 0.0309238962363452, 0.0310260178521276, 0.0320927088614553, 0.0335040374193341, 0.0340522730257362, 0.0344292330555618, 0.0348996436223388, 0.0359098089393228, 0.0366794092115015, 0.0381407886743546, 0.0384149765595794, 0.038526406744495, 0.0385718832258135, 0.0395206899847835, 0.0399406552314758, 0.0430012932047248, 0.0433284393511713, 0.0435569984838367, 0.0458493106998503, 0.046063638990745,
#> 0.0469350144267082, 0.0502011657226831, 0.0510624526068568, 0.0527009717188776, 0.056073043262586, 0.056558586191386, 0.0581756506580859, 0.0590829530265182, 0.0631373389624059, 0.0648830933496356, 0.065636639483273, 0.067319828318432, 0.0683417033869773, 0.0684064193628728, 0.0687954481691122, 0.0699455172289163, 0.0704171205870807, 0.0709647727198899, 0.0713426703587174, 0.0715497385244817, 0.0729944417253137, 0.0732091320678592, 0.0746336323209107, 0.07588304951787, 0.076176312752068, 0.0769833216909319,
#> 0.0770898594055325, 0.0783282660413533, 0.0783842522650957, 0.0791664109565318, 0.0796190109103918, 0.0808449615724385, 0.0818330235779285, 0.0844361330382526, 0.0852247001603246, 0.085870444541797, 0.0860389897134155, 0.087183809839189, 0.0877202423289418, 0.0898489013779908, 0.091200975002721, 0.091969795525074, 0.0923966427799314, 0.0939681925810874, 0.0940303672105074, 0.0946227721869946, 0.0959265041165054, 0.0979780545458198, 0.0985663856845349, 0.0990097080357373, 0.100656457711011, 0.101095365360379,
#> 0.101313760969788, 0.101950954878703, 0.104046371765435, 0.104306503897533, 0.106346960878, 0.107331892941147, 0.107689367607236, 0.107744531240314, 0.108949673594907, 0.109048785641789, 0.109670019010082, 0.110329848947003, 0.111680223839357, 0.111789910122752, 0.113047500839457, 0.114378662081435, 0.115189302247018, 0.116309177130461, 0.118044569622725, 0.118276783032343, 0.118683209177107, 0.120041428366676, 0.121108880266547, 0.12179663986899, 0.121864625485614, 0.124373451108113, 0.125789226964116,
#> 0.125856270780787, 0.126492621842772, 0.126666583819315, 0.127557650906965, 0.128115758299828, 0.128602578304708, 0.129016225924715, 0.130024369573221, 0.131418510107324, 0.133348547620699, 0.133507662685588, 0.135991170769557, 0.136628583772108, 0.137970441021025, 0.138059550896287, 0.138074840884656, 0.138926521642134, 0.139359317719936, 0.142943662358448, 0.143266438273713, 0.143559225834906, 0.143803449813277, 0.145115376915783, 0.145315843401477, 0.145407346775755, 0.148463048506528, 0.148710346780717,
#> 0.148782722884789, 0.150928949937224, 0.151495558675379, 0.155294531024992, 0.156360570108518, 0.159034826094285, 0.15922928112559, 0.159274747595191, 0.161409182474017, 0.162198703736067, 0.163412279449403, 0.163623198168352, 0.164675234118477, 0.165010989643633, 0.165283794514835, 0.166161776287481, 0.166490767849609, 0.166603677673265, 0.168132450431585, 0.168414602987468, 0.170263659209013, 0.170395507011563, 0.172351888613775, 0.173224841943011, 0.174386327620596, 0.174868220463395, 0.177759329089895,
#> 0.178613093914464, 0.179034290602431, 0.179127504583448, 0.18081600125879, 0.18237791932188, 0.18331205425784, 0.18332012812607, 0.185446366202086, 0.185653790598735, 0.190525120124221, 0.192943317815661, 0.192943505709991, 0.193312666844577, 0.194759515346959, 0.196219522040337, 0.197463317075744, 0.198986241826788, 0.199000779306516, 0.199160666903481, 0.200559966498986, 0.201238424051553, 0.202698964858428, 0.203208829043433, 0.203664392232895, 0.203900680411607, 0.204030646011233, 0.204329062951729,
#> 0.205179745331407, 0.205828856211156, 0.20680187526159, 0.20962158520706, 0.2101352927275, 0.210310738068074, 0.212967462604865, 0.21314931102097, 0.214942595688626, 0.215272658271715, 0.216099976096302, 0.216380230383947, 0.216625959379598, 0.216995889088139, 0.217905807076022, 0.218338430393487, 0.218562992988154, 0.219235972734168, 0.219439741224051, 0.22002470633015, 0.224765427643433, 0.225115388631821, 0.226108731469139, 0.226481329649687, 0.232457197736949, 0.237697737524286, 0.238379750866443,
#> 0.238453106489033, 0.23927381564863, 0.239288098411635, 0.239638329250738, 0.240461234701797, 0.24159372295253, 0.241776089882478, 0.241788821527734, 0.242454329272732, 0.24249267578125, 0.242587020387873, 0.243392595089972, 0.243791283341125, 0.244483520975336, 0.244927987456322, 0.246750867227092, 0.247077947948128, 0.247219110373408, 0.248734866967425, 0.250523259630427, 0.25157762831077, 0.252699653152376, 0.253311580745503, 0.254773765802383, 0.254818350775167, 0.255749259144068, 0.256982919061556,
#> 0.258426348445937, 0.25920274364762, 0.259509476367384, 0.259818243561313, 0.260058912448585, 0.261186117772013, 0.261237550061196, 0.261856944067404, 0.262909896671772, 0.26330864848569, 0.263841792242602, 0.264481923077255, 0.265187191078439, 0.265793564263731, 0.268330775666982, 0.269694633316249, 0.270977834472433, 0.272646195720881, 0.275805117562413, 0.276456848951057, 0.28040054673329, 0.281313059618697, 0.28133351309225, 0.281812581699342, 0.282959054922685, 0.285985574126244, 0.28618732560426,
#> 0.288237168220803, 0.288417027564719, 0.290648308349773, 0.291757629951462, 0.292968772351742, 0.293585412902758, 0.293750792043284, 0.293758839834481, 0.294625013135374, 0.295750094112009, 0.296028222655877, 0.297161926981062, 0.297560029197484, 0.298344632610679, 0.298658029874787, 0.301930186571553, 0.303833421086892, 0.30599033809267, 0.306281309342012, 0.307273737154901, 0.310959493741393, 0.312329404754564, 0.314662620658055, 0.315396910067648, 0.315594685263932, 0.315731404349208, 0.316115506459028,
#> 0.317790219327435, 0.318212727084756, 0.318756407825276, 0.318848039023578, 0.321854551089928, 0.326400522142649, 0.326429924694821, 0.327313161687925, 0.327338520204648, 0.328267297474667, 0.329317163676023, 0.329380251467228, 0.330931987147778, 0.330953701864928, 0.333823317429051, 0.33596885856241, 0.337348550790921, 0.338219850324094, 0.339253881014884, 0.340119243832305, 0.344429030548781, 0.345683183986694, 0.345853600883856, 0.349299048539251, 0.349695475772023, 0.350486536743119, 0.350588162429631,
#> 0.351248814957216, 0.352704691933468, 0.354186728131026, 0.355901510221884, 0.356314577395096, 0.359530670568347, 0.359960122732446, 0.360042401123792, 0.360777561552823, 0.360873693600297, 0.361006100894883, 0.361534116789699, 0.362516305875033, 0.363712143152952, 0.364262431394309, 0.365486512426287, 0.366688103647903, 0.367329021682963, 0.367464390583336, 0.368559058057144, 0.369236068567261, 0.369340825127438, 0.371269882190973, 0.373410821193829, 0.375140478601679, 0.375875194789842, 0.376467330846936,
#> 0.377343045547605, 0.378641178831458, 0.378960735630244, 0.379946742206812, 0.381056858925149, 0.381981546059251, 0.383370670489967, 0.385183100355789, 0.387096277438104, 0.387481981189921, 0.388349752407521, 0.38956403802149, 0.391453241696581, 0.391542107099667, 0.392751928884536, 0.393671740312129, 0.393777719466016, 0.394606579793617, 0.395447959192097, 0.395998113555834, 0.396137897623703, 0.396409955108538, 0.396673975745216, 0.397771646967158, 0.397993594175205, 0.398619815241545, 0.39930623723194,
#> 0.401598005788401, 0.401990393176675, 0.405270203948021, 0.405728721991181, 0.406697055324912, 0.406853629974648, 0.407361345365644, 0.407593891955912, 0.407973201479763, 0.408853879896924, 0.411679971730337, 0.412256387062371, 0.412472855532542, 0.414047828642651, 0.414248683257028, 0.415961450664327, 0.417264371411875, 0.417269926285371, 0.417966035660356, 0.419485852587968, 0.42137112445198, 0.423704410903156, 0.424597555538639, 0.42943161376752, 0.429831350920722, 0.43052065372467, 0.430898792576045,
#> 0.43100341851823, 0.431522403843701, 0.432788166450337, 0.432930769631639, 0.43359481333755, 0.435386642348021, 0.435718158725649, 0.437340782955289, 0.437655055196956, 0.438137200428173, 0.438757269177586, 0.438799632480368, 0.439149999525398, 0.440223303856328, 0.441499187611043, 0.442186920205131, 0.443733628606424, 0.444789632922038, 0.446997430175543, 0.449733827495947, 0.449776821769774, 0.451120329322293, 0.451336304657161, 0.451858439715579, 0.452552471542731, 0.453089235117659, 0.455840147798881,
#> 0.456749180797487, 0.458160649519414, 0.459017724497244, 0.459139019018039, 0.460747514152899, 0.46650480106473, 0.466509439516813, 0.466597515856847, 0.468089348869398, 0.468959034187719, 0.469878912670538, 0.47104003559798, 0.47175561147742, 0.473137095803395, 0.473980569280684, 0.475545125314966, 0.476504088379443, 0.477587363682687, 0.481484478106722, 0.483298178529367, 0.484119494911283, 0.484802823513746, 0.488288650754839, 0.490392812760547, 0.495559869799763, 0.495834674919024, 0.495889646699652,
#> 0.495991045143455, 0.496825250331312, 0.497852969681844, 0.501247070264071, 0.501423423178494, 0.502264236565679, 0.502747076796368, 0.503026663092896, 0.503146104514599, 0.503395081730559, 0.504012379329652, 0.508309862110764, 0.508893572259694, 0.510964883957058, 0.512136830715463, 0.513866862049326, 0.514402497094125, 0.514744562795386, 0.515355301322415, 0.515585619490594, 0.516034910222515, 0.516804101876915, 0.517485157353804, 0.517681184923276, 0.519060970284045, 0.521528884768486, 0.522895360132679,
#> 0.523428416578099, 0.523650436662138, 0.524280134355649, 0.525731408968568, 0.526106729870662, 0.526830323971808, 0.528322355356067, 0.528709272388369, 0.529934776481241, 0.530409487430006, 0.531741869403049, 0.532565164146945, 0.53384270472452, 0.535217588068917, 0.536133256973699, 0.538079284364358, 0.538192446576431, 0.538633120479062, 0.538760172203183, 0.540337681537494, 0.54043101449497, 0.542075810953975, 0.542837154818699, 0.543792691081762, 0.543831168906763, 0.544705166015774, 0.545425235992298,
#> 0.545723421266302, 0.545915438327938, 0.546367618953809, 0.546804774785414, 0.548591439146549, 0.5492207063362, 0.54946154775098, 0.550862395204604, 0.551290672272444, 0.552318777190521, 0.55247233598493, 0.552900205599144, 0.559383239364251, 0.559491197112948, 0.56438572704792, 0.56658627698198, 0.566909784451127, 0.567406318150461, 0.569740539882332, 0.570957343792543, 0.571560842683539, 0.57454437110573, 0.576187420636415, 0.578461039112881, 0.578630845760927, 0.578781491611153, 0.581366044003516,
#> 0.584308366058394, 0.586222191574052, 0.587346678134054, 0.588078625965863, 0.588247172767296, 0.589837556472048, 0.590643109753728, 0.59340836503543, 0.593836588552222, 0.594472281169146, 0.594819201389328, 0.595954202115536, 0.598474994301796, 0.600037338444963, 0.602003859123215, 0.602144769858569, 0.602222412126139, 0.603750402340665, 0.604173966916278, 0.605368442134932, 0.60564771364443, 0.606558852363378, 0.610299621941522, 0.612082786625251, 0.612376979552209, 0.613101419527084, 0.614005603594705,
#> 0.614524137461558, 0.614753181813285, 0.616014427971095, 0.616498636547476, 0.61691455822438, 0.618977224919945, 0.622798821423203, 0.622922778129578, 0.623061030171812, 0.626456494210288, 0.626966629410163, 0.62715786579065, 0.627365985419601, 0.62845016666688, 0.628682107897475, 0.629416507901624, 0.630411718972027, 0.630542930448428, 0.632092875195667, 0.633231563260779, 0.634520042454824, 0.634544272208586, 0.63566812267527, 0.636446596588939, 0.637685551773757, 0.638883732957765, 0.639060988556594,
#> 0.640680484939367, 0.641383435344324, 0.641941264504567, 0.643069031648338, 0.643738753627986, 0.644469760591164, 0.645054772496223, 0.646727143321186, 0.646781304851174, 0.64764387672767, 0.648082781815901, 0.648501313757151, 0.649506189860404, 0.650298700900748, 0.650343898683786, 0.650485265767202, 0.651118639623746, 0.652000183938071, 0.652479521231726, 0.652510039275512, 0.652967276517302, 0.653699397109449, 0.655982626136392, 0.65657958202064, 0.656856436515227, 0.656995829660445, 0.657298437319696,
#> 0.658322231844068, 0.661077789729461, 0.661573967430741, 0.664285492850468, 0.664530064212158, 0.665152461268008, 0.665890787029639, 0.666092011611909, 0.667815764434636, 0.668674502521753, 0.670548492344096, 0.671395304379985, 0.673447825945914, 0.673856367124245, 0.675023473566398, 0.675208503613248, 0.675661901943386, 0.676610797178, 0.677698523038998, 0.677978094667196, 0.677988059120253, 0.679920632624999, 0.680798615328968, 0.68464411306195, 0.686063104076311, 0.686513980152085, 0.686569210607558,
#> 0.690007234923542, 0.690405166242272, 0.690719101810828, 0.690725848078728, 0.690917830914259, 0.696488575544208, 0.696583657991141, 0.697655427269638, 0.700714985607192, 0.700767503585666, 0.701550335623324, 0.704463991336524, 0.70467192796059, 0.706139502348378, 0.706946498015895, 0.707500395365059, 0.708293445874006, 0.708484261762351, 0.708732047816738, 0.708912507165223, 0.710274052573368, 0.71097124973312, 0.71175341703929, 0.711955302860588, 0.713411951670423, 0.714371634181589, 0.715186134213582,
#> 0.715423076646402, 0.715809417888522, 0.719013958238065, 0.719317963346839, 0.719522785162553, 0.721405047690496, 0.721759729320183, 0.723450553836301, 0.72493509715423, 0.725763959344476, 0.726285317447037, 0.726948852883652, 0.727702046744525, 0.727905224077404, 0.72817997331731, 0.728305660188198, 0.729761208640411, 0.730338460532948, 0.73110364517197, 0.731229154625908, 0.732910666614771, 0.73297052201815, 0.733729686122388, 0.733746317680925, 0.734444106230512, 0.735729368403554, 0.737123666564003,
#> 0.737795152235776, 0.738091561477631, 0.73925079475157, 0.739589563803747, 0.739872505422682, 0.741845581447706, 0.742326180217788, 0.742623895406723, 0.742924000136554, 0.743018237408251, 0.743621315108612, 0.746071676257998, 0.747848246712238, 0.748737998073921, 0.749272652203217, 0.750239846063778, 0.750247884308919, 0.750603328226134, 0.751013845670968, 0.75395310902968, 0.754406645428389, 0.756041088141501, 0.758053619647399, 0.759158379863948, 0.759170953882858, 0.759677961235866, 0.76076113851741,
#> 0.760858229827136, 0.760893133934587, 0.761059405049309, 0.761255360208452, 0.762110914569348, 0.762819469673559, 0.763474826235324, 0.763934360351413, 0.76546558062546, 0.769013522192836, 0.769048036308959, 0.770643183263019, 0.772161534056067, 0.772408777615055, 0.772476847516373, 0.772730364464223, 0.773008481133729, 0.773027691291645, 0.774801583494991, 0.777306163217872, 0.777645506896079, 0.778292785864323, 0.778454303508624, 0.778970633633435, 0.781122568063438, 0.782154128421098, 0.783140670042485,
#> 0.783727515023202, 0.78417244553566, 0.784201819682494, 0.784709724131972, 0.785951164085418, 0.78942035860382, 0.791907660430297, 0.792102944571525, 0.792735469993204, 0.794477666728199, 0.795377714792266, 0.796250590356067, 0.800097748171538, 0.80127512710169, 0.803144115256146, 0.803412572713569, 0.805463733617216, 0.809108371380717, 0.809296958846971, 0.809696201002225, 0.814045199658722, 0.814231365453452, 0.814596203621477, 0.815023199422285, 0.815097741549835, 0.816187682561576, 0.817073642974719,
#> 0.817513144109398, 0.819082446629182, 0.82403287710622, 0.824915395118296, 0.825307863531634, 0.826700376812369, 0.827501414343715, 0.827921322081238, 0.828014417784289, 0.831581892445683, 0.83168991911225, 0.834508620202541, 0.836075305938721, 0.836446250556037, 0.837061725789681, 0.838375008897856, 0.838629989651963, 0.839265023823828, 0.841558910673484, 0.84309694217518, 0.843870285665616, 0.844163393368945, 0.844231416238472, 0.845540054840967, 0.845703554572538, 0.847479980904609, 0.847637428436428,
#> 0.849882774055004, 0.850553432479501, 0.850721327355132, 0.850974229164422, 0.851743174018338, 0.852432376472279, 0.852844803361222, 0.853635872947052, 0.853723727166653, 0.854173652129248, 0.854383887955919, 0.854959498858079, 0.855881966184825, 0.856081013800576, 0.856132156681269, 0.856522594578564, 0.858328149653971, 0.859285893617198, 0.859722247580066, 0.864375276723877, 0.864441073266789, 0.866615703329444, 0.869459297973663, 0.870324538787827, 0.870727153494954, 0.872587442398071, 0.872630304889753,
#> 0.873417352326214, 0.873739800183102, 0.874176835641265, 0.875357910292223, 0.875383010366932, 0.875410763313994, 0.875621909275651, 0.876767661655322, 0.880309032043442, 0.880933094071224, 0.880934217246249, 0.880956979934126, 0.881019168300554, 0.881312924902886, 0.882156537845731, 0.884369102073833, 0.884742884431034, 0.885090930387378, 0.885247568367049, 0.885660339612514, 0.885710798436776, 0.886374238179997, 0.886550510535017, 0.887723417486995, 0.887872519437224, 0.888015177566558, 0.890028999885544,
#> 0.891284866956994, 0.892745704157278, 0.894122657133266, 0.896378374658525, 0.897304410114884, 0.897902505705133, 0.898077371064574, 0.898567755473778, 0.902923798188567, 0.903877350268885, 0.906051577767357, 0.90618684887886, 0.906491699861363, 0.906879458576441, 0.909420361043885, 0.913158010691404, 0.913685443112627, 0.91429804963991, 0.916229565627873, 0.916653896914795, 0.918010637164116, 0.918337844777852, 0.920275803189725, 0.921526176389307, 0.921799641335383, 0.922474455786869, 0.922564618522301,
#> 0.923789556138217, 0.923960765823722, 0.924520392203704, 0.925656766165048, 0.926283334614709, 0.927281761541963, 0.927870102226734, 0.930233496706933, 0.930281535722315, 0.931795864831656, 0.933358375681564, 0.934172523673624, 0.934998403536156, 0.935517142061144, 0.936428893357515, 0.936487426050007, 0.937330218032002, 0.938152686925605, 0.942132205469534, 0.943459724308923, 0.943516464205459, 0.944774803239852, 0.945220490684733, 0.945334274321795, 0.94568784837611, 0.946294847643003, 0.946558905066922,
#> 0.947317183716223, 0.947318266145885, 0.947913724463433, 0.948578845243901, 0.94968924135901, 0.951395966811106, 0.952429423108697, 0.952809649286792, 0.952824799809605, 0.955866285832599, 0.957919124513865, 0.958503520581871, 0.958727898774669, 0.95948260743171, 0.961116276681423, 0.961571368388832, 0.962416304508224, 0.963181753642857, 0.963494028197601, 0.964343837695196, 0.964625613065436, 0.965119112050161, 0.965240417979658, 0.966159525094554, 0.966209195787087, 0.967139130458236, 0.968462549848482,
#> 0.968637502286583, 0.969707059208304, 0.969791890354827, 0.970262279734015, 0.970681362319738, 0.970845024799928, 0.97104093618691, 0.972844217903912, 0.973622557939962, 0.9748362575192, 0.975818040780723, 0.976875316584483, 0.97903377446346, 0.979293410433456, 0.97947422042489, 0.981531849130988, 0.981541850371286, 0.982190598966554, 0.982668666634709, 0.983266869559884, 0.983283746987581, 0.983511551516131, 0.983525407966226, 0.98456546664238, 0.986508697737008, 0.992580709746107, 0.993118094280362,
#> 0.994194674305618, 0.994832404656336, 0.995123026426882, 0.995962664484978, 0.996033379342407, 0.996412934735417, 0.999652457190678), N = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
#> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
#> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
#> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
#> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
#> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
#> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), cumN = 1:1000))