Mathematical and statistical functions for the Empirical distribution, which is commonly used in sampling such as MCMC.

## Value

Returns an R6 object inheriting from class SDistribution.

## Details

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).

## Distribution support

The distribution is supported on $$x_1,...,x_k$$.

Emp(samples = 1)

N/A

N/A

## References

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

## Super classes

distr6::Distribution -> distr6::SDistribution -> Empirical

## Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

alias

Alias of the distribution.

## Methods

Inherited methods

### Method new()

Creates a new instance of this R6 class.

### Method 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.

#### Arguments

which

(character(1) | numeric(1)
Ignored if distribution is unimodal. Otherwise "all" returns all modes, otherwise specifies which mode to return.

### Method 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.

#### Arguments

...

Unused.

### Method 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.

#### Arguments

base

(integer(1))
Base of the entropy logarithm, default = 2 (Shannon entropy)

...

Unused.

### Method 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.

#### Arguments

t

(integer(1))
t integer to evaluate function at.

...

Unused.

### Method 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.

Empirical$pgf(z, ...) #### Arguments z (integer(1)) z integer to evaluate probability generating function at. ... Unused. ### Method setParameterValue() Sets the value(s) of the given parameter(s). #### Usage Empirical$setParameterValue(
...,
lst = NULL,
error = "warn",
resolveConflicts = FALSE
)

#### Arguments

...

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 clone()

The objects of this class are cloneable with this method.

Empirical$clone(deep = FALSE) #### Arguments deep Whether to make a deep clone. ## Examples  ## ------------------------------------------------ ## Method Empirical$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))