Core ideas in Monte Carlo simulation
- modular arithmetic gives pseudo-random streams that are indistiguishable from 'true' Uniformly distributed samples in integers from \({0,1,2,...,m}\)
- by diving the integer streams from above by \(m\) we get samples from \({0/m,1/m,...,(m-1)/m}\) and "pretend" this to be samples from the Uniform(0,1) RV
- we can use inverse distribution function of von Neumann's rejection sampler to convert samples from Uniform(0,1) RV to the following:
- any other random variable
- vector of random variables that could be dependent
- or more generally other random structures:
- random graphs and networks
- random walks or (sensible perturbations of live traffic data on open street maps for hypothesis tests)
- models of interacting paticle systems in ecology / chemcal physics, etc...
- Source of randomness is the Standard Uniform RV on the Unit Interval: https://en.wikipedia.org/wiki/Continuousuniformdistribution
- Transform the Uniform RV using one of several methods, including:
- https://en.wikipedia.org/wiki/Inversetransformsampling
- https://en.wikipedia.org/wiki/Rejection_sampling - will revisit below for Expoential RV
breeze.stats.distributions
Breeze also provides a fairly large number of probability distributions. These come with access to probability density function for either discrete or continuous distributions. Many distributions also have methods for giving the mean and the variance.
Let us simulate from the Poisson distribution with the following probability mass and cumulative distribution functions:
import breeze.stats.distributions._
val poi = new Poisson(3.0);
import breeze.stats.distributions._
poi: breeze.stats.distributions.Poisson = Poisson(3.0)
val s = poi.sample(5); // let's draw five samples - black-box
s: IndexedSeq[Int] = Vector(3, 2, 5, 4, 1)
Getting probabilities of the Poisson samples
s.map( x => poi.probabilityOf(x) ) // PMF
res0: IndexedSeq[Double] = Vector(0.22404180765538775, 0.22404180765538775, 0.10081881344492458, 0.16803135574154085, 0.14936120510359185)
val doublePoi = for(x <- poi) yield x.toDouble // meanAndVariance requires doubles, but Poisson samples over Ints
doublePoi: breeze.stats.distributions.Rand[Double] = MappedRand(Poisson(3.0),$Lambda$8556/443731598@3fbbf491)
breeze.stats.meanAndVariance(doublePoi.samples.take(1000));
res1: breeze.stats.meanAndVariance.MeanAndVariance = MeanAndVariance(3.0660000000000034,3.250894894894892,1000)
(poi.mean, poi.variance) // population mean and variance
res1: (Double, Double) = (3.0,3.0)
Exponential random Variable
Let's focus on getting our hands dirty with the Exponential random variable, a common continuous RV with rate parameter \(\lambda\) with the following probability density and distribution functions:
NOTE: Below, there is a possibility of confusion for the term rate in the family of exponential distributions. Breeze parameterizes the distribution with the mean, but refers to it as the rate.
val expo = new Exponential(0.5);
expo: breeze.stats.distributions.Exponential = Exponential(0.5)
expo.rate // what is the rate parameter
res2: Double = 0.5
A characteristic of exponential distributions is its half-life, but we can compute the probability a value falls between any two numbers.
expo.probability(0, math.log(2) * expo.rate)
res3: Double = 0.1591035847462855
expo.probability(math.log(2) * expo.rate, 10000.0)
res4: Double = 0.8408964152537145
expo.probability(0.0, 1.5)
res5: Double = 0.5276334472589853
The above result means that approximately 95% of the draws from an exponential distribution fall between 0 and thrice the mean. We could have easily computed this with the cumulative distribution as well.
1 - math.exp(-3.0) // the CDF of the Exponential RV with rate parameter 3
res6: Double = 0.950212931632136
Drawing samples from Exponential RV
Using the Inverse Transform Sampling, we can sample from the Exponential RV by transforming the samples from the standard Uniform RV as shown in the animation below:
val samples = expo.sample(2).sorted; // built-in black box - we will roll our own shortly in Spark
samples: IndexedSeq[Double] = Vector(0.5783204633797516, 2.470656843546517)
expo.probability(samples(0), samples(1));
res7: Double = 0.4581529379432925
breeze.stats.meanAndVariance(expo.samples.take(10000)); // mean and variance of the sample
res8: breeze.stats.meanAndVariance.MeanAndVariance = MeanAndVariance(1.9763018410146713,3.9243880003317417,10000)
(1 / expo.rate, 1 / (expo.rate * expo.rate)) // mean and variance of the population
res9: (Double, Double) = (2.0,4.0)
import spark.implicits._
import org.apache.spark.sql.functions._
import spark.implicits._
import org.apache.spark.sql.functions._
val df = spark.range(1000).toDF("Id") // just make a DF of 1000 row indices
df: org.apache.spark.sql.DataFrame = [Id: bigint]
df.show(5)
+---+
| Id|
+---+
| 0|
| 1|
| 2|
| 3|
| 4|
+---+
only showing top 5 rows
val dfRand = df.select($"Id", rand(seed=1234567) as "rand") // add a column of random numbers in (0,1)
dfRand: org.apache.spark.sql.DataFrame = [Id: bigint, rand: double]
dfRand.show(5) // these are first 5 of the 1000 samples from the Uniform(0,1) RV
+---+--------------------+
| Id| rand|
+---+--------------------+
| 0|0.024042816995877625|
| 1| 0.4763832311243641|
| 2| 0.3392911406256911|
| 3| 0.6282132043405342|
| 4| 0.9665960987271114|
+---+--------------------+
only showing top 5 rows
val dfRand = df.select($"Id", rand(seed=1234567) as "rand") // add a column of random numbers in (0,1)
dfRand.show(5) // these are first 5 of the 1000 samples from the Uniform(0,1) RV
+---+--------------------+
| Id| rand|
+---+--------------------+
| 0|0.024042816995877625|
| 1| 0.4763832311243641|
| 2| 0.3392911406256911|
| 3| 0.6282132043405342|
| 4| 0.9665960987271114|
+---+--------------------+
only showing top 5 rows
dfRand: org.apache.spark.sql.DataFrame = [Id: bigint, rand: double]
val dfRand = df.select($"Id", rand(seed=879664) as "rand") // add a column of random numbers in (0,1)
dfRand.show(5) // these are first 5 of the 1000 samples from the Uniform(0,1) RV
+---+-------------------+
| Id| rand|
+---+-------------------+
| 0| 0.269224348263137|
| 1|0.20433965628039852|
| 2| 0.8481228617934538|
| 3| 0.6665103087537884|
| 4| 0.7712898780552342|
+---+-------------------+
only showing top 5 rows
dfRand: org.apache.spark.sql.DataFrame = [Id: bigint, rand: double]
Let's use the inverse CDF of the Exponential RV to transform these samples from the Uniform(0,1) RV into those from the Exponential RV.
val dfRand = df.select($"Id", rand(seed=1234567) as "rand") // add a column of random numbers in (0,1)
.withColumn("one",lit(1.0))
.withColumn("rate",lit(0.5))
dfRand: org.apache.spark.sql.DataFrame = [Id: bigint, rand: double ... 2 more fields]
dfRand.show(5)
+---+--------------------+---+----+
| Id| rand|one|rate|
+---+--------------------+---+----+
| 0|0.024042816995877625|1.0| 0.5|
| 1| 0.4763832311243641|1.0| 0.5|
| 2| 0.3392911406256911|1.0| 0.5|
| 3| 0.6282132043405342|1.0| 0.5|
| 4| 0.9665960987271114|1.0| 0.5|
+---+--------------------+---+----+
only showing top 5 rows
val dfExpRand = dfRand.withColumn("expo_sample", -($"one" / $"rate") * log($"one" - $"rand")) // samples from expo(rate=0.5)
dfExpRand: org.apache.spark.sql.DataFrame = [Id: bigint, rand: double ... 3 more fields]
dfExpRand.show(5)
+---+--------------------+---+----+--------------------+
| Id| rand|one|rate| expo_sample|
+---+--------------------+---+----+--------------------+
| 0|0.024042816995877625|1.0| 0.5|0.048673126808362034|
| 1| 0.4763832311243641|1.0| 0.5| 1.2939904386814756|
| 2| 0.3392911406256911|1.0| 0.5| 0.8288839819270684|
| 3| 0.6282132043405342|1.0| 0.5| 1.9788694379168241|
| 4| 0.9665960987271114|1.0| 0.5| 6.798165162486205|
+---+--------------------+---+----+--------------------+
only showing top 5 rows
display(dfExpRand)
| Id | rand | one | rate | expo_sample |
|---|---|---|---|---|
| 0.0 | 2.4042816995877625e-2 | 1.0 | 0.5 | 4.8673126808362034e-2 |
| 1.0 | 0.4763832311243641 | 1.0 | 0.5 | 1.2939904386814756 |
| 2.0 | 0.3392911406256911 | 1.0 | 0.5 | 0.8288839819270684 |
| 3.0 | 0.6282132043405342 | 1.0 | 0.5 | 1.9788694379168241 |
| 4.0 | 0.9665960987271114 | 1.0 | 0.5 | 6.798165162486205 |
| 5.0 | 0.8269758822163711 | 1.0 | 0.5 | 3.508648570093974 |
| 6.0 | 0.3404052214826948 | 1.0 | 0.5 | 0.8322592089262 |
| 7.0 | 0.5658253891225227 | 1.0 | 0.5 | 1.6686169931673005 |
| 8.0 | 0.11737981749647353 | 1.0 | 0.5 | 0.24972063061386013 |
| 9.0 | 0.7848668745585088 | 1.0 | 0.5 | 3.072996508744745 |
| 10.0 | 0.10665116151243736 | 1.0 | 0.5 | 0.2255562755600773 |
| 11.0 | 0.34971775733163646 | 1.0 | 0.5 | 0.8606975816973311 |
| 12.0 | 3.349566781262969e-2 | 1.0 | 0.5 | 6.813899599567436e-2 |
| 13.0 | 0.334063669089834 | 1.0 | 0.5 | 0.8131224244912618 |
| 14.0 | 0.4105052702588069 | 1.0 | 0.5 | 1.0569789985263547 |
| 15.0 | 0.22519777483698333 | 1.0 | 0.5 | 0.5102949510683874 |
| 16.0 | 0.21124107883891907 | 1.0 | 0.5 | 0.47458910937069004 |
| 17.0 | 0.12332274767843698 | 1.0 | 0.5 | 0.26323273531964136 |
| 18.0 | 4.324422155449681e-2 | 1.0 | 0.5 | 8.841423006745963e-2 |
| 19.0 | 3.933267172708754e-2 | 1.0 | 0.5 | 8.025420479220831e-2 |
| 20.0 | 0.8162723223794007 | 1.0 | 0.5 | 3.388601261211285 |
| 21.0 | 7.785566240785313e-2 | 1.0 | 0.5 | 0.16210703862675058 |
| 22.0 | 0.1244015150504949 | 1.0 | 0.5 | 0.26569528726942954 |
| 23.0 | 0.5313795676534989 | 1.0 | 0.5 | 1.5159243018004147 |
| 24.0 | 0.8177063965639969 | 1.0 | 0.5 | 3.4042733720631824 |
| 25.0 | 0.7757379079578416 | 1.0 | 0.5 | 2.98987971469377 |
| 26.0 | 0.9019086568714294 | 1.0 | 0.5 | 4.643712323362237 |
| 27.0 | 0.5793202308042869 | 1.0 | 0.5 | 1.7317667559523853 |
| 28.0 | 4.913530139386124e-2 | 1.0 | 0.5 | 0.10076699863506142 |
| 29.0 | 0.6984668419742928 | 1.0 | 0.5 | 2.3977505839877837 |
| 30.0 | 1.579141234703818e-2 | 1.0 | 0.5 | 3.18348501444197e-2 |
| 31.0 | 0.3170147319677016 | 1.0 | 0.5 | 0.762563978285606 |
| 32.0 | 0.6243818511105477 | 1.0 | 0.5 | 1.9583644261768265 |
| 33.0 | 0.9720896733059052 | 1.0 | 0.5 | 7.157517052464175 |
| 34.0 | 6.406755887221727e-2 | 1.0 | 0.5 | 0.13242396678361196 |
| 35.0 | 0.8275324400058236 | 1.0 | 0.5 | 3.515092236401242 |
| 36.0 | 0.4223760255739688 | 1.0 | 0.5 | 1.0976643705892708 |
| 37.0 | 0.6237792697270514 | 1.0 | 0.5 | 1.9551585184791915 |
| 38.0 | 0.30676209726152626 | 1.0 | 0.5 | 0.7327640894184466 |
| 39.0 | 0.7209798363387625 | 1.0 | 0.5 | 2.5529424571699533 |
| 40.0 | 0.5408086036735253 | 1.0 | 0.5 | 1.556576340750279 |
| 41.0 | 0.22254408000797787 | 1.0 | 0.5 | 0.5034566621599345 |
| 42.0 | 0.6478051949133747 | 1.0 | 0.5 | 2.0871416658496442 |
| 43.0 | 0.6186563924196967 | 1.0 | 0.5 | 1.9281089062362353 |
| 44.0 | 0.2689179760895458 | 1.0 | 0.5 | 0.6264592354306907 |
| 45.0 | 0.1310898372764776 | 1.0 | 0.5 | 0.28103107825164747 |
| 46.0 | 0.8459785027719904 | 1.0 | 0.5 | 3.741326187841892 |
| 47.0 | 0.7844461581951896 | 1.0 | 0.5 | 3.069089109365284 |
| 48.0 | 0.3195541476806115 | 1.0 | 0.5 | 0.7700140609818293 |
| 49.0 | 0.326892158146161 | 1.0 | 0.5 | 0.7916994433570004 |
| 50.0 | 0.4392312327086285 | 1.0 | 0.5 | 1.1568932758900772 |
| 51.0 | 0.5377790694946909 | 1.0 | 0.5 | 1.543424595293625 |
| 52.0 | 0.4369640582174875 | 1.0 | 0.5 | 1.1488236262486446 |
| 53.0 | 0.26672242876690055 | 1.0 | 0.5 | 0.6204619408451196 |
| 54.0 | 0.9743181848790359 | 1.0 | 0.5 | 7.323944240755433 |
| 55.0 | 0.5636877255447679 | 1.0 | 0.5 | 1.6587941323713102 |
| 56.0 | 0.8214194684040824 | 1.0 | 0.5 | 3.44543124420649 |
| 57.0 | 4.2394218638494574e-2 | 1.0 | 0.5 | 8.663817482333575e-2 |
| 58.0 | 0.2986030735782996 | 1.0 | 0.5 | 0.7093626466953578 |
| 59.0 | 0.9251464009715654 | 1.0 | 0.5 | 5.184442172120483 |
| 60.0 | 0.9768531076059933 | 1.0 | 0.5 | 7.531789490600966 |
| 61.0 | 0.4147651110232632 | 1.0 | 0.5 | 1.0714839854387161 |
| 62.0 | 0.4525161010109643 | 1.0 | 0.5 | 1.2048444519262322 |
| 63.0 | 7.638776464132979e-2 | 1.0 | 0.5 | 0.1589259082490955 |
| 64.0 | 0.8203900440907219 | 1.0 | 0.5 | 3.433935381714252 |
| 65.0 | 0.9746746584977457 | 1.0 | 0.5 | 7.35189948830076 |
| 66.0 | 0.41832172801299305 | 1.0 | 0.5 | 1.083675562745256 |
| 67.0 | 3.3131200470495226e-2 | 1.0 | 0.5 | 6.738494114620364e-2 |
| 68.0 | 0.10008247055930286 | 1.0 | 0.5 | 0.21090430762250917 |
| 69.0 | 0.4268021831375646 | 1.0 | 0.5 | 1.113048783438383 |
| 70.0 | 0.8213380254753332 | 1.0 | 0.5 | 3.444519337826204 |
| 71.0 | 0.47431121070064797 | 1.0 | 0.5 | 1.2860917933318652 |
| 72.0 | 0.68679164992865 | 1.0 | 0.5 | 2.321773309424188 |
| 73.0 | 0.7057803676609208 | 1.0 | 0.5 | 2.4468574835462356 |
| 74.0 | 0.32184620342731496 | 1.0 | 0.5 | 0.776762356325894 |
| 75.0 | 0.3589329148022541 | 1.0 | 0.5 | 0.8892423408857425 |
| 76.0 | 0.7448782338623747 | 1.0 | 0.5 | 2.7320286670643386 |
| 77.0 | 0.9740613004640858 | 1.0 | 0.5 | 7.304038469785621 |
| 78.0 | 0.7453909609063684 | 1.0 | 0.5 | 2.736052180743762 |
| 79.0 | 6.853610233988816e-2 | 1.0 | 0.5 | 0.14199569380051374 |
| 80.0 | 0.336012781812836 | 1.0 | 0.5 | 0.8189847588182159 |
| 81.0 | 0.2963058480260412 | 1.0 | 0.5 | 0.7028229208813994 |
| 82.0 | 0.38627232525200883 | 1.0 | 0.5 | 0.9764079513820425 |
| 83.0 | 0.9097248909817963 | 1.0 | 0.5 | 4.809787008391862 |
| 84.0 | 0.20679745942261907 | 1.0 | 0.5 | 0.46335335878970363 |
| 85.0 | 0.16582497569469312 | 1.0 | 0.5 | 0.36262407472768277 |
| 86.0 | 0.6960738838737711 | 1.0 | 0.5 | 2.381941292346302 |
| 87.0 | 0.890094879396298 | 1.0 | 0.5 | 4.416275650715409 |
| 88.0 | 0.8258738496535992 | 1.0 | 0.5 | 3.4959504809275685 |
| 89.0 | 0.4066244237870885 | 1.0 | 0.5 | 1.0438554620679272 |
| 90.0 | 0.44414816275414526 | 1.0 | 0.5 | 1.1745070000344822 |
| 91.0 | 0.8820223175378497 | 1.0 | 0.5 | 4.274519608161631 |
| 92.0 | 0.3814200182827251 | 1.0 | 0.5 | 0.9606575597598919 |
| 93.0 | 0.491252177347836 | 1.0 | 0.5 | 1.3516056440673538 |
| 94.0 | 0.9042145561359308 | 1.0 | 0.5 | 4.691289097027222 |
| 95.0 | 0.8293982814391448 | 1.0 | 0.5 | 3.536847140695551 |
| 96.0 | 0.6957817964411346 | 1.0 | 0.5 | 2.38002012037681 |
| 97.0 | 0.46108317902841456 | 1.0 | 0.5 | 1.2363880819986401 |
| 98.0 | 0.658234439306857 | 1.0 | 0.5 | 2.14726054405596 |
| 99.0 | 0.43175037735005384 | 1.0 | 0.5 | 1.130388960612226 |
| 100.0 | 0.7719881477151886 | 1.0 | 0.5 | 2.9567153353469786 |
| 101.0 | 6.397418621476536e-2 | 1.0 | 0.5 | 0.13222444810891013 |
| 102.0 | 0.44410081132020296 | 1.0 | 0.5 | 1.1743366329905425 |
| 103.0 | 0.9364231799229901 | 1.0 | 0.5 | 5.511012678534471 |
| 104.0 | 0.5453471566845608 | 1.0 | 0.5 | 1.576442265948382 |
| 105.0 | 0.7346238968987211 | 1.0 | 0.5 | 2.653214404406468 |
| 106.0 | 0.3493365535175731 | 1.0 | 0.5 | 0.8595254994862054 |
| 107.0 | 0.5230037795263933 | 1.0 | 0.5 | 1.480493423321206 |
| 108.0 | 0.6886814284788837 | 1.0 | 0.5 | 2.333877090719847 |
| 109.0 | 0.7731252946239237 | 1.0 | 0.5 | 2.966714745163421 |
| 110.0 | 2.276168830750991e-2 | 1.0 | 0.5 | 4.604946957472576e-2 |
| 111.0 | 0.264951110753621 | 1.0 | 0.5 | 0.6156365319998442 |
| 112.0 | 0.14616140102686104 | 1.0 | 0.5 | 0.3160261944637388 |
| 113.0 | 3.066619523108849e-2 | 1.0 | 0.5 | 6.229248529326732e-2 |
| 114.0 | 0.49017052188807597 | 1.0 | 0.5 | 1.3473579316333943 |
| 115.0 | 9.835381148010536e-2 | 1.0 | 0.5 | 0.20706617613153588 |
| 116.0 | 8.828156810764243e-2 | 1.0 | 0.5 | 0.1848481470740138 |
| 117.0 | 0.6868446103815881 | 1.0 | 0.5 | 2.3221115183574854 |
| 118.0 | 0.4067862578001439 | 1.0 | 0.5 | 1.0444010055396156 |
| 119.0 | 0.44070042328022496 | 1.0 | 0.5 | 1.1621400679168603 |
| 120.0 | 0.19523663294276805 | 1.0 | 0.5 | 0.4344139974853618 |
| 121.0 | 0.34930320794183467 | 1.0 | 0.5 | 0.8594230049585315 |
| 122.0 | 0.37713232784782635 | 1.0 | 0.5 | 0.9468423740115679 |
| 123.0 | 0.6429635193876022 | 1.0 | 0.5 | 2.0598346316676848 |
| 124.0 | 0.5726782020473655 | 1.0 | 0.5 | 1.7004358488091253 |
| 125.0 | 0.3013872989669776 | 1.0 | 0.5 | 0.7173175321607891 |
| 126.0 | 0.3469657736553403 | 1.0 | 0.5 | 0.8522514741503977 |
| 127.0 | 0.3062597218624077 | 1.0 | 0.5 | 0.7313152550300903 |
| 128.0 | 0.875187814442521 | 1.0 | 0.5 | 4.161890374256847 |
| 129.0 | 0.4331447487608374 | 1.0 | 0.5 | 1.1353025933318837 |
| 130.0 | 0.40669672646934574 | 1.0 | 0.5 | 1.0440991764725864 |
| 131.0 | 0.10105613827873039 | 1.0 | 0.5 | 0.21306938341833667 |
| 132.0 | 0.7598058191143243 | 1.0 | 0.5 | 2.852615191501923 |
| 133.0 | 0.7882552857444408 | 1.0 | 0.5 | 3.104747815909758 |
| 134.0 | 0.41196255767852374 | 1.0 | 0.5 | 1.0619293113867083 |
| 135.0 | 0.23956793737837967 | 1.0 | 0.5 | 0.5477370075782519 |
| 136.0 | 0.301117813053034 | 1.0 | 0.5 | 0.716546192187808 |
| 137.0 | 0.4359866364410945 | 1.0 | 0.5 | 1.1453546670228079 |
| 138.0 | 0.9682651399507317 | 1.0 | 0.5 | 6.90067903242789 |
| 139.0 | 0.10678335714273168 | 1.0 | 0.5 | 0.22585225268919112 |
| 140.0 | 0.2693226978676214 | 1.0 | 0.5 | 0.6275667277003328 |
| 141.0 | 0.5668302775070247 | 1.0 | 0.5 | 1.6732513179468083 |
| 142.0 | 0.7096151948853401 | 1.0 | 0.5 | 2.473096642771549 |
| 143.0 | 0.7574375459472653 | 1.0 | 0.5 | 2.8329921185549365 |
| 144.0 | 0.49769652658104957 | 1.0 | 0.5 | 1.3771016264425564 |
| 145.0 | 0.45884572648841415 | 1.0 | 0.5 | 1.2281017543594028 |
| 146.0 | 0.3288386747446713 | 1.0 | 0.5 | 0.7974914915764556 |
| 147.0 | 0.5410136911709724 | 1.0 | 0.5 | 1.557469795251325 |
| 148.0 | 0.6336220553800798 | 1.0 | 0.5 | 2.008179685617141 |
| 149.0 | 0.375647544465191 | 1.0 | 0.5 | 0.9420804749655175 |
| 150.0 | 0.10142418148506294 | 1.0 | 0.5 | 0.21388838577034947 |
| 151.0 | 0.4721728293756773 | 1.0 | 0.5 | 1.2779727544450452 |
| 152.0 | 0.7518331384524283 | 1.0 | 0.5 | 2.787307860488268 |
| 153.0 | 6.676079335730689e-2 | 1.0 | 0.5 | 0.13818745319668635 |
| 154.0 | 0.8053497028664404 | 1.0 | 0.5 | 3.2731013568502365 |
| 155.0 | 0.6436250826649516 | 1.0 | 0.5 | 2.063543927419815 |
| 156.0 | 0.28432165233043427 | 1.0 | 0.5 | 0.6690488961123614 |
| 157.0 | 0.33131329522029473 | 1.0 | 0.5 | 0.8048792646192418 |
| 158.0 | 0.4390246975576103 | 1.0 | 0.5 | 1.1561567971907922 |
| 159.0 | 0.9576167619712271 | 1.0 | 0.5 | 6.322004648835221 |
| 160.0 | 0.2191897706358752 | 1.0 | 0.5 | 0.4948462856734829 |
| 161.0 | 1.1385964425825512e-2 | 1.0 | 0.5 | 2.2902561570486413e-2 |
| 162.0 | 0.679111841181116 | 1.0 | 0.5 | 2.2733252629142697 |
| 163.0 | 0.3189329784940914 | 1.0 | 0.5 | 0.768189122733912 |
| 164.0 | 0.5494024842158357 | 1.0 | 0.5 | 1.5943615282559738 |
| 165.0 | 0.4995339525325778 | 1.0 | 0.5 | 1.3844310395116766 |
| 166.0 | 0.6133797107105055 | 1.0 | 0.5 | 1.900624464472158 |
| 167.0 | 0.18020899363866738 | 1.0 | 0.5 | 0.3974116829996968 |
| 168.0 | 0.3549472573452819 | 1.0 | 0.5 | 0.8768463879435472 |
| 169.0 | 0.6992620601216817 | 1.0 | 0.5 | 2.403032050173198 |
| 170.0 | 0.24085612119427668 | 1.0 | 0.5 | 0.5511279118150403 |
| 171.0 | 0.1609790935708849 | 1.0 | 0.5 | 0.3510393091093769 |
| 172.0 | 0.9711366383834527 | 1.0 | 0.5 | 7.090364504579701 |
| 173.0 | 0.38455734034855193 | 1.0 | 0.5 | 0.9708269965922631 |
| 174.0 | 0.7140149247000265 | 1.0 | 0.5 | 2.503631307579515 |
| 175.0 | 0.6231707003505403 | 1.0 | 0.5 | 1.9519259602967969 |
| 176.0 | 0.6056033420465197 | 1.0 | 0.5 | 1.8607962600766825 |
| 177.0 | 0.9191984318132235 | 1.0 | 0.5 | 5.03151781078248 |
| 178.0 | 0.31549310135748787 | 1.0 | 0.5 | 0.7581131118739614 |
| 179.0 | 0.4450415722231543 | 1.0 | 0.5 | 1.1777241458955992 |
| 180.0 | 0.4583485174404076 | 1.0 | 0.5 | 1.2262650109539877 |
| 181.0 | 0.8948048795370758 | 1.0 | 0.5 | 4.503876726373103 |
| 182.0 | 0.5068846132533013 | 1.0 | 0.5 | 1.4140241642575553 |
| 183.0 | 0.474099338611806 | 1.0 | 0.5 | 1.2852858815130643 |
| 184.0 | 3.968996187382612e-2 | 1.0 | 0.5 | 8.099818055602134e-2 |
| 185.0 | 0.3812923011104965 | 1.0 | 0.5 | 0.9602446657347301 |
| 186.0 | 0.4490455490374048 | 1.0 | 0.5 | 1.1922062787516934 |
| 187.0 | 0.18885603597795153 | 1.0 | 0.5 | 0.4186194528265905 |
| 188.0 | 0.14835262533228888 | 1.0 | 0.5 | 0.32116543464514347 |
| 189.0 | 1.0239721160210324e-2 | 1.0 | 0.5 | 2.0585015521643695e-2 |
| 190.0 | 0.7812863611722196 | 1.0 | 0.5 | 3.0399839801249593 |
| 191.0 | 0.40315501297127665 | 1.0 | 0.5 | 1.032195705046974 |
| 192.0 | 0.27623054102229216 | 1.0 | 0.5 | 0.6465647282627558 |
| 193.0 | 0.3476188563715197 | 1.0 | 0.5 | 0.8542526234724834 |
| 194.0 | 0.40005917714351136 | 1.0 | 0.5 | 1.0218485144052543 |
| 195.0 | 0.690531564629077 | 1.0 | 0.5 | 2.3457983558717395 |
| 196.0 | 0.6243976540207664 | 1.0 | 0.5 | 1.9584485714328679 |
| 197.0 | 0.6936108435709636 | 1.0 | 0.5 | 2.365798463973219 |
| 198.0 | 0.5580754032112615 | 1.0 | 0.5 | 1.6332320139161962 |
| 199.0 | 0.5077201201493545 | 1.0 | 0.5 | 1.4174157254902722 |
| 200.0 | 0.24217287516689856 | 1.0 | 0.5 | 0.5545999737072358 |
| 201.0 | 0.5866156019146013 | 1.0 | 0.5 | 1.7667547458541368 |
| 202.0 | 0.13480908234777678 | 1.0 | 0.5 | 0.289610164710414 |
| 203.0 | 0.8592340571612042 | 1.0 | 0.5 | 3.921313495527709 |
| 204.0 | 0.5015919278028333 | 1.0 | 0.5 | 1.3926722308356132 |
| 205.0 | 0.9524596166650006 | 1.0 | 0.5 | 6.092351507324665 |
| 206.0 | 0.7897861372493181 | 1.0 | 0.5 | 3.1192597448504493 |
| 207.0 | 0.992502482003315 | 1.0 | 0.5 | 9.786366493971752 |
| 208.0 | 0.7190312712917515 | 1.0 | 0.5 | 2.539023803153757 |
| 209.0 | 4.905914294075553e-2 | 1.0 | 0.5 | 0.10060681726863402 |
| 210.0 | 7.292405751548614e-2 | 1.0 | 0.5 | 0.15143958783801956 |
| 211.0 | 0.7448249635342986 | 1.0 | 0.5 | 2.731611103575813 |
| 212.0 | 7.024810923770186e-2 | 1.0 | 0.5 | 0.14567502510922456 |
| 213.0 | 0.6499895525785409 | 1.0 | 0.5 | 2.0995845503371515 |
| 214.0 | 0.2347304081263114 | 1.0 | 0.5 | 0.5350541991171558 |
| 215.0 | 0.5554391604460018 | 1.0 | 0.5 | 1.62133672301354 |
| 216.0 | 0.45655849903065526 | 1.0 | 0.5 | 1.2196664242497346 |
| 217.0 | 0.7734426024060143 | 1.0 | 0.5 | 2.969513910302441 |
| 218.0 | 0.13396935067038884 | 1.0 | 0.5 | 0.28766995842079884 |
| 219.0 | 0.44732092861928796 | 1.0 | 0.5 | 1.1859555740131458 |
| 220.0 | 0.532360601482588 | 1.0 | 0.5 | 1.5201155921058849 |
| 221.0 | 0.8353842261129955 | 1.0 | 0.5 | 3.6082823273928657 |
| 222.0 | 0.4415319449291001 | 1.0 | 0.5 | 1.1651157196663489 |
| 223.0 | 7.107670903097285e-2 | 1.0 | 0.5 | 0.14745823038641734 |
| 224.0 | 0.2587810776672368 | 1.0 | 0.5 | 0.5989185111514972 |
| 225.0 | 0.8783986898097359 | 1.0 | 0.5 | 4.214015069834256 |
| 226.0 | 0.8772054215086009 | 1.0 | 0.5 | 4.194484826673072 |
| 227.0 | 0.8792794678306639 | 1.0 | 0.5 | 4.228554112475122 |
| 228.0 | 0.9704733519907857 | 1.0 | 0.5 | 7.04492420208042 |
| 229.0 | 0.5911949898493493 | 1.0 | 0.5 | 1.7890339688352386 |
| 230.0 | 0.1881415659970821 | 1.0 | 0.5 | 0.4168585927615873 |
| 231.0 | 0.27280409460696464 | 1.0 | 0.5 | 0.6371187335647226 |
| 232.0 | 0.8997405595900834 | 1.0 | 0.5 | 4.599988097103155 |
| 233.0 | 0.805703299381296 | 1.0 | 0.5 | 3.2767378072078768 |
| 234.0 | 0.12469160466325713 | 1.0 | 0.5 | 0.26635800581530467 |
| 235.0 | 4.5007500538482015e-2 | 1.0 | 0.5 | 9.210358499849247e-2 |
| 236.0 | 0.5406049480217902 | 1.0 | 0.5 | 1.5556895188057671 |
| 237.0 | 0.7089850749112344 | 1.0 | 0.5 | 2.4687614483220965 |
| 238.0 | 0.4050350886413143 | 1.0 | 0.5 | 1.038505695363729 |
| 239.0 | 0.18681323647475456 | 1.0 | 0.5 | 0.4135889487659769 |
| 240.0 | 0.23557258393827696 | 1.0 | 0.5 | 0.5372564022794455 |
| 241.0 | 0.32077303736600393 | 1.0 | 0.5 | 0.7735998942749642 |
| 242.0 | 0.27136665743377386 | 1.0 | 0.5 | 0.6331692658855377 |
| 243.0 | 8.600287997937284e-2 | 1.0 | 0.5 | 0.1798557169901321 |
| 244.0 | 0.7600332087019175 | 1.0 | 0.5 | 2.8545094696108473 |
| 245.0 | 0.41783858737032487 | 1.0 | 0.5 | 1.0820150568291664 |
| 246.0 | 0.8293413803867148 | 1.0 | 0.5 | 3.5361801888515507 |
| 247.0 | 0.7333405374926266 | 1.0 | 0.5 | 2.6435657118891944 |
| 248.0 | 0.8772160342203914 | 1.0 | 0.5 | 4.194657687243259 |
| 249.0 | 5.301719603632349e-2 | 1.0 | 0.5 | 0.10894868878842578 |
| 250.0 | 0.9026197915984311 | 1.0 | 0.5 | 4.658264576287863 |
| 251.0 | 0.23914978744357662 | 1.0 | 0.5 | 0.5466375404979962 |
| 252.0 | 0.4968410398019236 | 1.0 | 0.5 | 1.3736982691137887 |
| 253.0 | 0.5384099353933083 | 1.0 | 0.5 | 1.5461561756710942 |
| 254.0 | 0.8907377100572391 | 1.0 | 0.5 | 4.428007915156274 |
| 255.0 | 0.9066080833593035 | 1.0 | 0.5 | 4.741900966190897 |
| 256.0 | 0.6829340520390256 | 1.0 | 0.5 | 2.297290978019848 |
| 257.0 | 0.6571961620790489 | 1.0 | 0.5 | 2.1411937930382208 |
| 258.0 | 0.3669337202760652 | 1.0 | 0.5 | 0.9143603100291875 |
| 259.0 | 0.5097789017273943 | 1.0 | 0.5 | 1.4257975373650602 |
| 260.0 | 0.19359708351354943 | 1.0 | 0.5 | 0.4303435299938143 |
| 261.0 | 0.4198903553742862 | 1.0 | 0.5 | 1.0890763016996372 |
| 262.0 | 0.8275841161941834 | 1.0 | 0.5 | 3.515691583104315 |
| 263.0 | 0.5245014108343904 | 1.0 | 0.5 | 1.486782728111085 |
| 264.0 | 0.20212413144380958 | 1.0 | 0.5 | 0.45160449363942295 |
| 265.0 | 0.6588746438800428 | 1.0 | 0.5 | 2.1510105119936225 |
| 266.0 | 0.6090021282726809 | 1.0 | 0.5 | 1.8781063243284435 |
| 267.0 | 0.8237249377890233 | 1.0 | 0.5 | 3.4714193009141487 |
| 268.0 | 0.39616901908028135 | 1.0 | 0.5 | 1.0089219062444619 |
| 269.0 | 0.46974976524528655 | 1.0 | 0.5 | 1.268812485625708 |
| 270.0 | 0.1430991547459871 | 1.0 | 0.5 | 0.30886613379677225 |
| 271.0 | 0.8900411063855386 | 1.0 | 0.5 | 4.415297354889731 |
| 272.0 | 0.8908538606807918 | 1.0 | 0.5 | 4.430135134053363 |
| 273.0 | 0.7571966906626759 | 1.0 | 0.5 | 2.8310071799990957 |
| 274.0 | 0.45373865486682496 | 1.0 | 0.5 | 1.2093155273309903 |
| 275.0 | 0.5105750301805768 | 1.0 | 0.5 | 1.429048215969818 |
| 276.0 | 0.8703161512938904 | 1.0 | 0.5 | 4.085311447017532 |
| 277.0 | 0.6263910497078812 | 1.0 | 0.5 | 1.9690912320594496 |
| 278.0 | 0.8879229935650624 | 1.0 | 0.5 | 4.377138172983162 |
| 279.0 | 0.4592765782031528 | 1.0 | 0.5 | 1.2296947319737286 |
| 280.0 | 0.3332564208219061 | 1.0 | 0.5 | 0.8106994919909759 |
| 281.0 | 0.7769330816977306 | 1.0 | 0.5 | 3.000566940929362 |
| 282.0 | 0.47688504943059673 | 1.0 | 0.5 | 1.2959080966079761 |
| 283.0 | 0.5519970113210297 | 1.0 | 0.5 | 1.6059107508619763 |
| 284.0 | 0.6116679748262741 | 1.0 | 0.5 | 1.8917891406149387 |
| 285.0 | 0.23344992857646873 | 1.0 | 0.5 | 0.5317105161007432 |
| 286.0 | 0.547367711693101 | 1.0 | 0.5 | 1.5853504174370916 |
| 287.0 | 0.45229676089796134 | 1.0 | 0.5 | 1.2040433464044202 |
| 288.0 | 0.2232542399926749 | 1.0 | 0.5 | 0.5052843787124497 |
| 289.0 | 4.812719615823391e-2 | 1.0 | 0.5 | 9.864772505442727e-2 |
| 290.0 | 0.9924754840764293 | 1.0 | 0.5 | 9.779177599019986 |
| 291.0 | 0.23112957413067814 | 1.0 | 0.5 | 0.525665641185302 |
| 292.0 | 0.7136348699646448 | 1.0 | 0.5 | 2.500975207955907 |
| 293.0 | 0.7097521146725535 | 1.0 | 0.5 | 2.474039888248458 |
| 294.0 | 0.2534528100999812 | 1.0 | 0.5 | 0.584592898262904 |
| 295.0 | 0.14159162922836588 | 1.0 | 0.5 | 0.30535067223059725 |
| 296.0 | 0.286451779155965 | 1.0 | 0.5 | 0.6750105216507689 |
| 297.0 | 0.9194163349443866 | 1.0 | 0.5 | 5.0369186336261 |
| 298.0 | 0.8565261729771216 | 1.0 | 0.5 | 3.8832053010040535 |
| 299.0 | 0.9963446427954837 | 1.0 | 0.5 | 11.223122920363782 |
| 300.0 | 0.44312606197302895 | 1.0 | 0.5 | 1.1708327755623118 |
| 301.0 | 0.3333780003343948 | 1.0 | 0.5 | 0.8110642217087809 |
| 302.0 | 8.100442373331274e-2 | 1.0 | 0.5 | 0.16894794055205103 |
| 303.0 | 0.9114279295358697 | 1.0 | 0.5 | 4.847877405697818 |
| 304.0 | 0.1420851500734669 | 1.0 | 0.5 | 0.30650085385781334 |
| 305.0 | 0.3998938086649432 | 1.0 | 0.5 | 1.0212973077353178 |
| 306.0 | 0.43509561875462044 | 1.0 | 0.5 | 1.1421975977833791 |
| 307.0 | 0.2607501670030833 | 1.0 | 0.5 | 0.6042386923167598 |
| 308.0 | 0.7861629300585208 | 1.0 | 0.5 | 3.0850818187043214 |
| 309.0 | 0.4238639225351366 | 1.0 | 0.5 | 1.1028228011782435 |
| 310.0 | 0.8296205577621936 | 1.0 | 0.5 | 3.5394546320186806 |
| 311.0 | 0.797111503088613 | 1.0 | 0.5 | 3.190197454292849 |
| 312.0 | 4.847906767173793e-2 | 1.0 | 0.5 | 9.93871863877833e-2 |
| 313.0 | 0.2272307227002427 | 1.0 | 0.5 | 0.5155495038419591 |
| 314.0 | 0.28798275880438495 | 1.0 | 0.5 | 0.6793063054019258 |
| 315.0 | 0.9119630135350312 | 1.0 | 0.5 | 4.859996504134525 |
| 316.0 | 0.9247506337150422 | 1.0 | 0.5 | 5.173895593701935 |
| 317.0 | 0.313064117705499 | 1.0 | 0.5 | 0.7510286422175019 |
| 318.0 | 0.10267561230063371 | 1.0 | 0.5 | 0.2166756921335689 |
| 319.0 | 0.7800056454356652 | 1.0 | 0.5 | 3.0283067880604637 |
| 320.0 | 0.9880505590690435 | 1.0 | 0.5 | 8.854141571438799 |
| 321.0 | 0.5199897620784429 | 1.0 | 0.5 | 1.4678956926088331 |
| 322.0 | 0.9631286266437556 | 1.0 | 0.5 | 6.6006396376399366 |
| 323.0 | 0.6526508389996869 | 1.0 | 0.5 | 2.1148495545527686 |
| 324.0 | 0.44331946385558707 | 1.0 | 0.5 | 1.1715274946406817 |
| 325.0 | 0.7657175467075138 | 1.0 | 0.5 | 2.9024556523792184 |
| 326.0 | 0.8587810198761535 | 1.0 | 0.5 | 3.914887085632034 |
| 327.0 | 0.17717500338441594 | 1.0 | 0.5 | 0.39002348344727783 |
| 328.0 | 0.6400146315652561 | 1.0 | 0.5 | 2.043383783189525 |
| 329.0 | 0.5429950331288752 | 1.0 | 0.5 | 1.5661220394400488 |
| 330.0 | 0.1519419896098827 | 1.0 | 0.5 | 0.3296124741021507 |
| 331.0 | 0.2317505808062127 | 1.0 | 0.5 | 0.5272816679676412 |
| 332.0 | 0.9587236527128663 | 1.0 | 0.5 | 6.374931295970532 |
| 333.0 | 0.3957876337227971 | 1.0 | 0.5 | 1.0076590860947048 |
| 334.0 | 0.7520930392631228 | 1.0 | 0.5 | 2.7894035230513716 |
| 335.0 | 0.5219702075466132 | 1.0 | 0.5 | 1.4761644422492128 |
| 336.0 | 0.2718748554726389 | 1.0 | 0.5 | 0.6345646874711566 |
| 337.0 | 0.43339022569610974 | 1.0 | 0.5 | 1.1361688818668743 |
| 338.0 | 0.654264591604089 | 1.0 | 0.5 | 2.124163024174615 |
| 339.0 | 0.21929573754620202 | 1.0 | 0.5 | 0.4951177321724933 |
| 340.0 | 0.6463525037251117 | 1.0 | 0.5 | 2.0789092703893584 |
| 341.0 | 0.40363690799123597 | 1.0 | 0.5 | 1.0338111652659434 |
| 342.0 | 0.5464987531537895 | 1.0 | 0.5 | 1.5815145200909522 |
| 343.0 | 0.2794928368714429 | 1.0 | 0.5 | 0.6555998434128101 |
| 344.0 | 0.9679988160115437 | 1.0 | 0.5 | 6.883964754455241 |
| 345.0 | 0.9374750976583118 | 1.0 | 0.5 | 5.5443807282558 |
| 346.0 | 0.5043382898350185 | 1.0 | 0.5 | 1.403723241814482 |
| 347.0 | 0.37940066231801983 | 1.0 | 0.5 | 0.9541391883810038 |
| 348.0 | 0.16327499528946765 | 1.0 | 0.5 | 0.35651962241911506 |
| 349.0 | 0.6814882638496493 | 1.0 | 0.5 | 2.2881919129060675 |
| 350.0 | 0.7593963453274154 | 1.0 | 0.5 | 2.8492085714581132 |
| 351.0 | 8.2716537875273e-2 | 1.0 | 0.5 | 0.17267747098244784 |
| 352.0 | 0.9864630812678177 | 1.0 | 0.5 | 8.604669210362268 |
| 353.0 | 0.10362777838201431 | 1.0 | 0.5 | 0.2187990527194605 |
| 354.0 | 0.3018065167087475 | 1.0 | 0.5 | 0.7185180358786611 |
| 355.0 | 0.1981827289480086 | 1.0 | 0.5 | 0.4417490773130338 |
| 356.0 | 0.2519352721846343 | 1.0 | 0.5 | 0.5805315404780675 |
| 357.0 | 0.37720491978724313 | 1.0 | 0.5 | 0.947075477038406 |
| 358.0 | 3.334532342207808e-2 | 1.0 | 0.5 | 6.782791058537331e-2 |
| 359.0 | 0.49142274606532255 | 1.0 | 0.5 | 1.3522762997820723 |
| 360.0 | 0.7658697744126436 | 1.0 | 0.5 | 2.9037555976399063 |
| 361.0 | 0.430584327777621 | 1.0 | 0.5 | 1.1262891608345273 |
| 362.0 | 0.18774246417057594 | 1.0 | 0.5 | 0.41587565350859623 |
| 363.0 | 0.43213770208564395 | 1.0 | 0.5 | 1.131752645804099 |
| 364.0 | 0.5254311249185892 | 1.0 | 0.5 | 1.4906970370036838 |
| 365.0 | 0.21632961088151903 | 1.0 | 0.5 | 0.48753353815321687 |
| 366.0 | 0.5866314734106438 | 1.0 | 0.5 | 1.766831535403086 |
| 367.0 | 0.2127018826329221 | 1.0 | 0.5 | 0.47829660009371444 |
| 368.0 | 0.762880957964817 | 1.0 | 0.5 | 2.8783859537652954 |
| 369.0 | 0.7092177159493649 | 1.0 | 0.5 | 2.4703609132000657 |
| 370.0 | 0.4668447454243504 | 1.0 | 0.5 | 1.25788522569854 |
| 371.0 | 8.984189541128584e-2 | 1.0 | 0.5 | 0.18827390651246967 |
| 372.0 | 0.8401841637115331 | 1.0 | 0.5 | 3.6674662997626886 |
| 373.0 | 0.1769541258232804 | 1.0 | 0.5 | 0.3894866793361383 |
| 374.0 | 0.32759234424821526 | 1.0 | 0.5 | 0.793780983603854 |
| 375.0 | 0.7549436630804378 | 1.0 | 0.5 | 2.812534296521244 |
| 376.0 | 0.5406091390043264 | 1.0 | 0.5 | 1.5557077645471398 |
| 377.0 | 7.226167164454322e-2 | 1.0 | 0.5 | 0.15001111942581233 |
| 378.0 | 3.9174989741246e-2 | 1.0 | 0.5 | 7.992595578900699e-2 |
| 379.0 | 0.43530038855865494 | 1.0 | 0.5 | 1.1429227007667369 |
| 380.0 | 0.19425051459560916 | 1.0 | 0.5 | 0.4319647938819565 |
| 381.0 | 0.2690512431721227 | 1.0 | 0.5 | 0.6268238435762967 |
| 382.0 | 0.6122531354155542 | 1.0 | 0.5 | 1.894805126272543 |
| 383.0 | 0.8144904991435312 | 1.0 | 0.5 | 3.3692983613822673 |
| 384.0 | 0.44435200950498954 | 1.0 | 0.5 | 1.1752405917023965 |
| 385.0 | 0.40334065577595324 | 1.0 | 0.5 | 1.0328178822819034 |
| 386.0 | 0.6712465016160815 | 1.0 | 0.5 | 2.2248941081495746 |
| 387.0 | 0.610470870145168 | 1.0 | 0.5 | 1.8856332573010395 |
| 388.0 | 0.3753417849616416 | 1.0 | 0.5 | 0.941101269529212 |
| 389.0 | 0.5760552709957159 | 1.0 | 0.5 | 1.7163043767384967 |
| 390.0 | 0.8958499208801377 | 1.0 | 0.5 | 4.523844703206438 |
| 391.0 | 0.6375933030298327 | 1.0 | 0.5 | 2.0299764509716054 |
| 392.0 | 0.4996734547364764 | 1.0 | 0.5 | 1.3849886064074164 |
| 393.0 | 0.43886175915693537 | 1.0 | 0.5 | 1.1555759704008648 |
| 394.0 | 2.1262190388847357e-2 | 1.0 | 0.5 | 4.298297362707392e-2 |
| 395.0 | 0.2612612193574123 | 1.0 | 0.5 | 0.6056217946491596 |
| 396.0 | 0.9053544127651678 | 1.0 | 0.5 | 4.715232048676575 |
| 397.0 | 5.715293130681975e-3 | 1.0 | 0.5 | 1.146337583128797e-2 |
| 398.0 | 0.2947471489241186 | 1.0 | 0.5 | 0.6983977729250015 |
| 399.0 | 8.460986179878727e-2 | 1.0 | 0.5 | 0.17680984806639513 |
| 400.0 | 0.30069220409765574 | 1.0 | 0.5 | 0.7153285923659113 |
| 401.0 | 0.997746334354857 | 1.0 | 0.5 | 12.190394425627364 |
| 402.0 | 0.18308241899251243 | 1.0 | 0.5 | 0.40443413850132504 |
| 403.0 | 0.3105228234689785 | 1.0 | 0.5 | 0.7436433675489958 |
| 404.0 | 0.9179013483983409 | 1.0 | 0.5 | 4.999667373023489 |
| 405.0 | 0.11926274283498617 | 1.0 | 0.5 | 0.2539918600578725 |
| 406.0 | 0.43776630003690786 | 1.0 | 0.5 | 1.1516753585822221 |
| 407.0 | 0.4762249664302428 | 1.0 | 0.5 | 1.2933860241922412 |
| 408.0 | 0.8482098508688078 | 1.0 | 0.5 | 3.770512619720418 |
| 409.0 | 0.4567392533274949 | 1.0 | 0.5 | 1.2203317557106708 |
| 410.0 | 0.4435608409662263 | 1.0 | 0.5 | 1.1723948842501484 |
| 411.0 | 0.50408147116895 | 1.0 | 0.5 | 1.4026872442755594 |
| 412.0 | 0.36104698819880265 | 1.0 | 0.5 | 0.8958487225315878 |
| 413.0 | 0.4194713065185254 | 1.0 | 0.5 | 1.0876321003142069 |
| 414.0 | 7.486133824518293e-2 | 1.0 | 0.5 | 0.1556232962089628 |
| 415.0 | 3.373382017468085e-2 | 1.0 | 0.5 | 6.863186850542395e-2 |
| 416.0 | 0.4932147269781292 | 1.0 | 0.5 | 1.3593357794290557 |
| 417.0 | 0.8616586394474591 | 1.0 | 0.5 | 3.956062042024912 |
| 418.0 | 0.4117184281160471 | 1.0 | 0.5 | 1.0610991639078917 |
| 419.0 | 9.116459643129304e-2 | 1.0 | 0.5 | 0.19118255076481538 |
| 420.0 | 0.5890322019489482 | 1.0 | 0.5 | 1.7784808355923856 |
| 421.0 | 0.43778596825837024 | 1.0 | 0.5 | 1.1517453243828897 |
| 422.0 | 0.5541973559190224 | 1.0 | 0.5 | 1.6157578539111792 |
| 423.0 | 0.5236632829365838 | 1.0 | 0.5 | 1.4832605720870944 |
| 424.0 | 0.6204546477584022 | 1.0 | 0.5 | 1.9375623680773675 |
| 425.0 | 0.970316394356556 | 1.0 | 0.5 | 7.034320768236008 |
| 426.0 | 0.43993687089356626 | 1.0 | 0.5 | 1.1594115421186992 |
| 427.0 | 0.8405879195529187 | 1.0 | 0.5 | 3.6725254570044403 |
| 428.0 | 0.749280638128116 | 1.0 | 0.5 | 2.766842091120008 |
| 429.0 | 0.5488427514547121 | 1.0 | 0.5 | 1.5918786677038552 |
| 430.0 | 0.13747140435906435 | 1.0 | 0.5 | 0.29577395251982774 |
| 431.0 | 0.4576343120245375 | 1.0 | 0.5 | 1.2236296079783713 |
| 432.0 | 0.8148665411500688 | 1.0 | 0.5 | 3.3733566295966297 |
| 433.0 | 0.19547202902357552 | 1.0 | 0.5 | 0.4349990900088173 |
| 434.0 | 0.4890095869756548 | 1.0 | 0.5 | 1.3428089003184602 |
| 435.0 | 0.12715136653541503 | 1.0 | 0.5 | 0.27198624962848195 |
| 436.0 | 0.2032311331013058 | 1.0 | 0.5 | 0.45438129228481317 |
| 437.0 | 0.8787527473448604 | 1.0 | 0.5 | 4.21984681587113 |
| 438.0 | 0.4663010675042861 | 1.0 | 0.5 | 1.2558467916792808 |
| 439.0 | 0.3142004342512348 | 1.0 | 0.5 | 0.7543397443148356 |
| 440.0 | 0.5376680285543044 | 1.0 | 0.5 | 1.5429441860463613 |
| 441.0 | 0.8288823435429962 | 1.0 | 0.5 | 3.530807819241641 |
| 442.0 | 0.5381362365823195 | 1.0 | 0.5 | 1.5449706315293206 |
| 443.0 | 0.8741054855972965 | 1.0 | 0.5 | 4.144621819895548 |
| 444.0 | 0.9650736097727945 | 1.0 | 0.5 | 6.709025137110639 |
| 445.0 | 8.401495565683514e-2 | 1.0 | 0.5 | 0.17551048315515766 |
| 446.0 | 0.16014272925069362 | 1.0 | 0.5 | 0.34904663471346736 |
| 447.0 | 0.6761260538348454 | 1.0 | 0.5 | 2.254801787874353 |
| 448.0 | 0.30491746315485757 | 1.0 | 0.5 | 0.7274493648359186 |
| 449.0 | 0.8823536465498564 | 1.0 | 0.5 | 4.280144318376048 |
| 450.0 | 0.35138375129442345 | 1.0 | 0.5 | 0.8658280669121068 |
| 451.0 | 0.8162562408450885 | 1.0 | 0.5 | 3.388426210497536 |
| 452.0 | 0.673617698352461 | 1.0 | 0.5 | 2.2393717605117374 |
| 453.0 | 8.256976107323988e-2 | 1.0 | 0.5 | 0.1723574716228483 |
| 454.0 | 0.8365768515057421 | 1.0 | 0.5 | 3.6228248778772634 |
| 455.0 | 0.48971308441437955 | 1.0 | 0.5 | 1.3455642637453662 |
| 456.0 | 8.774182190296687e-2 | 1.0 | 0.5 | 0.18366447790309529 |
| 457.0 | 0.24099238723353444 | 1.0 | 0.5 | 0.5514869432829278 |
| 458.0 | 0.5907862348164701 | 1.0 | 0.5 | 1.787035212429029 |
| 459.0 | 0.9282830732565702 | 1.0 | 0.5 | 5.270056963848408 |
| 460.0 | 0.32874248016649343 | 1.0 | 0.5 | 0.7972048609806637 |
| 461.0 | 0.7759823021398251 | 1.0 | 0.5 | 2.9920604438874734 |
| 462.0 | 0.9906880008613337 | 1.0 | 0.5 | 9.352902960980467 |
| 463.0 | 0.35519401347986057 | 1.0 | 0.5 | 0.8776116070547805 |
| 464.0 | 0.22036578800170115 | 1.0 | 0.5 | 0.4978608565413814 |
| 465.0 | 0.8023870524533302 | 1.0 | 0.5 | 3.242889943586732 |
| 466.0 | 0.6555700243159146 | 1.0 | 0.5 | 2.131728945611721 |
| 467.0 | 0.6966117625572236 | 1.0 | 0.5 | 2.385483963919468 |
| 468.0 | 0.31199672354331665 | 1.0 | 0.5 | 0.7479233575368842 |
| 469.0 | 0.10874684309305482 | 1.0 | 0.5 | 0.23025353029665035 |
| 470.0 | 0.7768279542152332 | 1.0 | 0.5 | 2.999624598436637 |
| 471.0 | 0.5194659966765868 | 1.0 | 0.5 | 1.465714573067321 |
| 472.0 | 3.387321287945699e-2 | 1.0 | 0.5 | 6.892040754981558e-2 |
| 473.0 | 0.9137616097231142 | 1.0 | 0.5 | 4.901279675232361 |
| 474.0 | 0.9310800753319032 | 1.0 | 0.5 | 5.349619920714008 |
| 475.0 | 0.8956753753595035 | 1.0 | 0.5 | 4.520495700986587 |
| 476.0 | 0.9734569283996768 | 1.0 | 0.5 | 7.257973044045264 |
| 477.0 | 0.3100378260015375 | 1.0 | 0.5 | 0.7422370063712049 |
| 478.0 | 0.9004067891568347 | 1.0 | 0.5 | 4.613322561880238 |
| 479.0 | 0.3595142946826074 | 1.0 | 0.5 | 0.8910569518000615 |
| 480.0 | 0.5858555389814262 | 1.0 | 0.5 | 1.7630808527274318 |
| 481.0 | 0.7207027332090898 | 1.0 | 0.5 | 2.5509571840094765 |
| 482.0 | 0.5207773901843563 | 1.0 | 0.5 | 1.4711801017479904 |
| 483.0 | 0.5010641338355079 | 1.0 | 0.5 | 1.3905554324221692 |
| 484.0 | 0.7386177898402442 | 1.0 | 0.5 | 2.683543072229971 |
| 485.0 | 0.7514865091276058 | 1.0 | 0.5 | 2.784516291388268 |
| 486.0 | 0.996791280204512 | 1.0 | 0.5 | 11.48376647798819 |
| 487.0 | 0.31973896223026976 | 1.0 | 0.5 | 0.7705573508031445 |
| 488.0 | 0.5409498257000886 | 1.0 | 0.5 | 1.557191525390985 |
| 489.0 | 0.2351915468213719 | 1.0 | 0.5 | 0.5362597290194078 |
| 490.0 | 0.5722823786456511 | 1.0 | 0.5 | 1.6985841286612482 |
| 491.0 | 0.8872393295593034 | 1.0 | 0.5 | 4.364975334044577 |
| 492.0 | 0.7466377111677412 | 1.0 | 0.5 | 2.745869685758634 |
| 493.0 | 0.8944045014489479 | 1.0 | 0.5 | 4.496279071952017 |
| 494.0 | 0.7761279146229962 | 1.0 | 0.5 | 2.993360875321733 |
| 495.0 | 6.282087728304298e-4 | 1.0 | 0.5 | 1.2568123572811955e-3 |
| 496.0 | 2.1755806290161828e-2 | 1.0 | 0.5 | 4.399190658802096e-2 |
| 497.0 | 0.25822043228050306 | 1.0 | 0.5 | 0.5974063169929631 |
| 498.0 | 0.36706789505445514 | 1.0 | 0.5 | 0.9147842435207847 |
| 499.0 | 0.8535706465084604 | 1.0 | 0.5 | 3.8424243911227456 |
| 500.0 | 0.9583607379261688 | 1.0 | 0.5 | 6.357423513924509 |
| 501.0 | 0.1736332737734787 | 1.0 | 0.5 | 0.38143325101594394 |
| 502.0 | 0.7191057722209122 | 1.0 | 0.5 | 2.539554188214649 |
| 503.0 | 0.5866772207662699 | 1.0 | 0.5 | 1.7670528869782172 |
| 504.0 | 0.44342504955089734 | 1.0 | 0.5 | 1.171906870972165 |
| 505.0 | 0.8638825079861409 | 1.0 | 0.5 | 3.988473708673089 |
| 506.0 | 0.19206484030065873 | 1.0 | 0.5 | 0.42654694315586256 |
| 507.0 | 6.329398370441452e-2 | 1.0 | 0.5 | 0.1307715918497558 |
| 508.0 | 0.8324305296152746 | 1.0 | 0.5 | 3.5727145302718077 |
| 509.0 | 0.788341172013531 | 1.0 | 0.5 | 3.105559205156182 |
| 510.0 | 0.9968069620813552 | 1.0 | 0.5 | 11.493564979531474 |
| 511.0 | 0.5738675981141458 | 1.0 | 0.5 | 1.70601035690584 |
| 512.0 | 2.1924901743684888e-2 | 1.0 | 0.5 | 4.433764862438196e-2 |
| 513.0 | 0.31475857342097047 | 1.0 | 0.5 | 0.755968110508673 |
| 514.0 | 0.5840924030390661 | 1.0 | 0.5 | 1.7545843321676697 |
| 515.0 | 0.7770937124350712 | 1.0 | 0.5 | 3.0020076619607927 |
| 516.0 | 0.7130309607190969 | 1.0 | 0.5 | 2.496761892223379 |
| 517.0 | 0.3991683439699586 | 1.0 | 0.5 | 1.0188809802465286 |
| 518.0 | 0.48509787688675254 | 1.0 | 0.5 | 1.3275568971757934 |
| 519.0 | 0.45169086393350866 | 1.0 | 0.5 | 1.2018320683571915 |
| 520.0 | 0.12902719463817103 | 1.0 | 0.5 | 0.2762890498682689 |
| 521.0 | 0.2592134704552379 | 1.0 | 0.5 | 0.6000855589507411 |
| 522.0 | 0.2813001484016546 | 1.0 | 0.5 | 0.660622921987986 |
| 523.0 | 0.3762566475369137 | 1.0 | 0.5 | 0.9440325786940167 |
| 524.0 | 0.9267694189625785 | 1.0 | 0.5 | 5.228284343045507 |
| 525.0 | 0.15197347800388006 | 1.0 | 0.5 | 0.32968673548099053 |
| 526.0 | 0.14593597548755854 | 1.0 | 0.5 | 0.3154982356972103 |
| 527.0 | 0.35991105880863383 | 1.0 | 0.5 | 0.8922962833448616 |
| 528.0 | 0.3071108301936344 | 1.0 | 0.5 | 0.7337704414228866 |
| 529.0 | 0.18406488587857806 | 1.0 | 0.5 | 0.40684088837547255 |
| 530.0 | 0.5136711150110314 | 1.0 | 0.5 | 1.4417403317355608 |
| 531.0 | 0.6056599490894785 | 1.0 | 0.5 | 1.8610833370818218 |
| 532.0 | 0.44302266378392197 | 1.0 | 0.5 | 1.1704614578045645 |
| 533.0 | 0.7453946147476727 | 1.0 | 0.5 | 2.736080882533261 |
| 534.0 | 0.35502683979054106 | 1.0 | 0.5 | 0.8770931502610587 |
| 535.0 | 0.8797839338461947 | 1.0 | 0.5 | 4.236929207935352 |
| 536.0 | 6.238712029644411e-2 | 1.0 | 0.5 | 0.12883624673709432 |
| 537.0 | 0.9424758304767943 | 1.0 | 0.5 | 5.711100159925468 |
| 538.0 | 0.5704103694439081 | 1.0 | 0.5 | 1.689849747036119 |
| 539.0 | 0.9158448954522559 | 1.0 | 0.5 | 4.950187400162322 |
| 540.0 | 0.8873811703242918 | 1.0 | 0.5 | 4.367492702015337 |
| 541.0 | 0.6216382152888731 | 1.0 | 0.5 | 1.9438088773652653 |
| 542.0 | 0.48827370203072595 | 1.0 | 0.5 | 1.3399307423143256 |
| 543.0 | 0.4572473011375716 | 1.0 | 0.5 | 1.2222029953319737 |
| 544.0 | 3.967393021768029e-2 | 1.0 | 0.5 | 8.096479233410842e-2 |
| 545.0 | 0.29505351687188286 | 1.0 | 0.5 | 0.6992667790155687 |
| 546.0 | 0.5639878272499856 | 1.0 | 0.5 | 1.6601702337428519 |
| 547.0 | 0.48965142848521204 | 1.0 | 0.5 | 1.3453226263340796 |
| 548.0 | 0.8749613569725964 | 1.0 | 0.5 | 4.15826489047167 |
| 549.0 | 0.9256239598413039 | 1.0 | 0.5 | 5.19724285977426 |
| 550.0 | 0.7569605924580269 | 1.0 | 0.5 | 2.8290633556705096 |
| 551.0 | 0.3844172120051691 | 1.0 | 0.5 | 0.9703716742615802 |
| 552.0 | 0.1362858511639865 | 1.0 | 0.5 | 0.29302682234878386 |
| 553.0 | 0.37983323803435165 | 1.0 | 0.5 | 0.9555337323937093 |
| 554.0 | 0.26566735487098103 | 1.0 | 0.5 | 0.6175863160615908 |
| 555.0 | 0.36961293921650573 | 1.0 | 0.5 | 0.9228425321120206 |
| 556.0 | 9.63913755811967e-2 | 1.0 | 0.5 | 0.2027178998486325 |
| 557.0 | 0.18788147800222665 | 1.0 | 0.5 | 0.4162179728411676 |
| 558.0 | 0.26504942094508055 | 1.0 | 0.5 | 0.6159040428220617 |
| 559.0 | 0.739578089934357 | 1.0 | 0.5 | 2.6909044643005315 |
| 560.0 | 0.6457524310661318 | 1.0 | 0.5 | 2.075518526013794 |
| 561.0 | 0.9092071659989257 | 1.0 | 0.5 | 4.798349834364611 |
| 562.0 | 0.210969342425946 | 1.0 | 0.5 | 0.4739002053005348 |
| 563.0 | 0.9875371965277739 | 1.0 | 0.5 | 8.770013586320161 |
| 564.0 | 5.026001823606596e-2 | 1.0 | 0.5 | 0.10313407051509839 |
| 565.0 | 0.12100530799389475 | 1.0 | 0.5 | 0.2579528399771316 |
| 566.0 | 0.5890018173984741 | 1.0 | 0.5 | 1.7783329727795534 |
| 567.0 | 0.7224532694248171 | 1.0 | 0.5 | 2.563531923007902 |
| 568.0 | 0.8883622267112601 | 1.0 | 0.5 | 4.384991631947751 |
| 569.0 | 1.997234871111797e-2 | 1.0 | 0.5 | 4.0348984229344e-2 |
| 570.0 | 0.2212019516532856 | 1.0 | 0.5 | 0.5000070229243507 |
| 571.0 | 0.4273320266042193 | 1.0 | 0.5 | 1.1148983665533385 |
| 572.0 | 0.9981606891187609 | 1.0 | 0.5 | 12.596728597154938 |
| 573.0 | 0.24173972097342367 | 1.0 | 0.5 | 0.5534571525106269 |
| 574.0 | 0.30996923292210077 | 1.0 | 0.5 | 0.7420381848339302 |
| 575.0 | 0.5586292644735815 | 1.0 | 0.5 | 1.635740173145147 |
| 576.0 | 0.5710479605847417 | 1.0 | 0.5 | 1.6928203250772282 |
| 577.0 | 0.7716668116970227 | 1.0 | 0.5 | 2.953898729115331 |
| 578.0 | 0.1347753271289409 | 1.0 | 0.5 | 0.2895321367059855 |
| 579.0 | 0.7338594799945437 | 1.0 | 0.5 | 2.647461677978153 |
| 580.0 | 0.11653928663743118 | 1.0 | 0.5 | 0.2478169105193333 |
| 581.0 | 0.939970706008818 | 1.0 | 0.5 | 5.6258452054414265 |
| 582.0 | 0.6523955046706064 | 1.0 | 0.5 | 2.1133799063928125 |
| 583.0 | 0.8032152138335183 | 1.0 | 0.5 | 3.2512892068337553 |
| 584.0 | 0.6707576127357098 | 1.0 | 0.5 | 2.22192212014176 |
| 585.0 | 0.706492182007378 | 1.0 | 0.5 | 2.451702005911791 |
| 586.0 | 0.7647771000645677 | 1.0 | 0.5 | 2.894443408052288 |
| 587.0 | 0.8173666289040515 | 1.0 | 0.5 | 3.400549144677044 |
| 588.0 | 0.21872300084295448 | 1.0 | 0.5 | 0.49365103921530656 |
| 589.0 | 0.7310001086753175 | 1.0 | 0.5 | 2.626088606755685 |
| 590.0 | 0.8697372961374548 | 1.0 | 0.5 | 4.076404137303028 |
| 591.0 | 0.16400157324713815 | 1.0 | 0.5 | 0.35825709554754664 |
| 592.0 | 0.443313209241608 | 1.0 | 0.5 | 1.1715050236598006 |
| 593.0 | 0.8053657385685586 | 1.0 | 0.5 | 3.2732661278566844 |
| 594.0 | 0.30079618388942353 | 1.0 | 0.5 | 0.7156259936633084 |
| 595.0 | 0.45365879994299696 | 1.0 | 0.5 | 1.2090231797639022 |
| 596.0 | 0.8069809893677921 | 1.0 | 0.5 | 3.2899331884862204 |
| 597.0 | 0.8245884511775761 | 1.0 | 0.5 | 3.4812407168767363 |
| 598.0 | 0.41234621682526684 | 1.0 | 0.5 | 1.063234617242673 |
| 599.0 | 0.7968250346184004 | 1.0 | 0.5 | 3.1873755454599793 |
| 600.0 | 0.3927417451741644 | 1.0 | 0.5 | 0.9976022348148476 |
| 601.0 | 0.2728256265562098 | 1.0 | 0.5 | 0.6371779535572102 |
| 602.0 | 8.203703710226784e-2 | 1.0 | 0.5 | 0.1711964692057045 |
| 603.0 | 0.37776869105901545 | 1.0 | 0.5 | 0.9488867520931888 |
| 604.0 | 0.629411417011517 | 1.0 | 0.5 | 1.9853255448725573 |
| 605.0 | 0.7501910378551001 | 1.0 | 0.5 | 2.774117609305619 |
| 606.0 | 0.5996333900313177 | 1.0 | 0.5 | 1.8307492534099181 |
| 607.0 | 6.970492186242305e-2 | 1.0 | 0.5 | 0.14450690968030883 |
| 608.0 | 0.8940723635978487 | 1.0 | 0.5 | 4.489998186900931 |
| 609.0 | 0.6914380476986998 | 1.0 | 0.5 | 2.3516652759309613 |
| 610.0 | 0.3650550811756098 | 1.0 | 0.5 | 0.9084340517211708 |
| 611.0 | 6.128389132423373e-2 | 1.0 | 0.5 | 0.12648435832908939 |
| 612.0 | 0.7680345331232672 | 1.0 | 0.5 | 2.9223335361285874 |
| 613.0 | 0.23058140936561866 | 1.0 | 0.5 | 0.5242402528938145 |
| 614.0 | 0.18527936511039267 | 1.0 | 0.5 | 0.40982000756014936 |
| 615.0 | 0.28710359124883345 | 1.0 | 0.5 | 0.676838316784801 |
| 616.0 | 0.2108511727487865 | 1.0 | 0.5 | 0.4736006964588179 |
| 617.0 | 0.4624966724507077 | 1.0 | 0.5 | 1.241640656420202 |
| 618.0 | 0.9811961480020145 | 1.0 | 0.5 | 7.947387073248349 |
| 619.0 | 0.8245811182215507 | 1.0 | 0.5 | 3.4811571100355176 |
| 620.0 | 0.8168888824501829 | 1.0 | 0.5 | 3.3953242213722077 |
| 621.0 | 0.6208021859846258 | 1.0 | 0.5 | 1.9393945466905502 |
| 622.0 | 0.6514815007857316 | 1.0 | 0.5 | 2.108127935592991 |
| 623.0 | 0.8002695403068458 | 1.0 | 0.5 | 3.221573045869622 |
| 624.0 | 0.44461502006124476 | 1.0 | 0.5 | 1.176187496315193 |
| 625.0 | 0.24944701714221795 | 1.0 | 0.5 | 0.5738900673091492 |
| 626.0 | 0.15074825367641687 | 1.0 | 0.5 | 0.32679923126166255 |
| 627.0 | 0.48159464793185214 | 1.0 | 0.5 | 1.3139956195836684 |
| 628.0 | 0.7293846186875164 | 1.0 | 0.5 | 2.614113446702638 |
| 629.0 | 0.22027782974542087 | 1.0 | 0.5 | 0.49763522946247984 |
| 630.0 | 0.20798244662851184 | 1.0 | 0.5 | 0.46634344813095197 |
| 631.0 | 0.6919106866218325 | 1.0 | 0.5 | 2.354731119087316 |
| 632.0 | 0.9720879966592072 | 1.0 | 0.5 | 7.1573969108149464 |
| 633.0 | 0.9555225762947214 | 1.0 | 0.5 | 6.2255471020245 |
| 634.0 | 0.9266176357074408 | 1.0 | 0.5 | 5.22414328144902 |
| 635.0 | 0.3345233531787639 | 1.0 | 0.5 | 0.8145034658807747 |
| 636.0 | 0.718429616730576 | 1.0 | 0.5 | 2.5347456665564865 |
| 637.0 | 0.8878295863454977 | 1.0 | 0.5 | 4.375472027171012 |
| 638.0 | 0.3092083755044568 | 1.0 | 0.5 | 0.7398341142762942 |
| 639.0 | 0.7848160471899103 | 1.0 | 0.5 | 3.0725240444024844 |
| 640.0 | 0.4053789118552892 | 1.0 | 0.5 | 1.0396618058885414 |
| 641.0 | 0.7411589342475445 | 1.0 | 0.5 | 2.7030821027672483 |
| 642.0 | 0.45881127046875336 | 1.0 | 0.5 | 1.2279744157261765 |
| 643.0 | 0.22682375665456067 | 1.0 | 0.5 | 0.5144965144587033 |
| 644.0 | 0.16776368155499566 | 1.0 | 0.5 | 0.3672776837982627 |
| 645.0 | 0.5154869010668474 | 1.0 | 0.5 | 1.449221624070085 |
| 646.0 | 0.34454745235530915 | 1.0 | 0.5 | 0.8448587389647154 |
| 647.0 | 0.7272962306454143 | 1.0 | 0.5 | 2.5987383337541456 |
| 648.0 | 0.9326344100560882 | 1.0 | 0.5 | 5.395241852776035 |
| 649.0 | 0.29777872274027883 | 1.0 | 0.5 | 0.7070134297056129 |
| 650.0 | 0.9174126215561729 | 1.0 | 0.5 | 4.987796826948644 |
| 651.0 | 4.5376575536940744e-2 | 1.0 | 0.5 | 9.28766723998495e-2 |
| 652.0 | 0.4810458793989424 | 1.0 | 0.5 | 1.3118795986726604 |
| 653.0 | 0.2950958156948137 | 1.0 | 0.5 | 0.6993867883865484 |
| 654.0 | 0.2212183176044329 | 1.0 | 0.5 | 0.5000490521080256 |
| 655.0 | 0.8206110378697538 | 1.0 | 0.5 | 3.436397715675071 |
| 656.0 | 0.5629205916000906 | 1.0 | 0.5 | 1.6552807756173478 |
| 657.0 | 0.6994457934362445 | 1.0 | 0.5 | 2.4042543067509254 |
| 658.0 | 0.9830737705685453 | 1.0 | 0.5 | 8.15778164570834 |
| 659.0 | 0.8325491011982105 | 1.0 | 0.5 | 3.5741302243464643 |
| 660.0 | 0.2556320014497683 | 1.0 | 0.5 | 0.5904394894534792 |
| 661.0 | 0.5735719811333211 | 1.0 | 0.5 | 1.7046233960032402 |
| 662.0 | 0.3675775710486384 | 1.0 | 0.5 | 0.916395415751471 |
| 663.0 | 0.34003954250576196 | 1.0 | 0.5 | 0.8311507172880653 |
| 664.0 | 0.11876686109163148 | 1.0 | 0.5 | 0.2528661163846483 |
| 665.0 | 0.4006303818268926 | 1.0 | 0.5 | 1.0237536248988528 |
| 666.0 | 0.6505245015357007 | 1.0 | 0.5 | 2.10264364860553 |
| 667.0 | 0.10086413330967303 | 1.0 | 0.5 | 0.21264225003430867 |
| 668.0 | 0.5732670526062165 | 1.0 | 0.5 | 1.7031937546945186 |
| 669.0 | 0.11878163120957308 | 1.0 | 0.5 | 0.25289963814199007 |
| 670.0 | 0.10072655638961314 | 1.0 | 0.5 | 0.21233625313017732 |
| 671.0 | 0.5618155656093103 | 1.0 | 0.5 | 1.6502307483050531 |
| 672.0 | 0.4500194541428453 | 1.0 | 0.5 | 1.1957447451000092 |
| 673.0 | 0.3981179573374246 | 1.0 | 0.5 | 1.0153875905870227 |
| 674.0 | 0.3056380605496083 | 1.0 | 0.5 | 0.7295238556749573 |
| 675.0 | 0.30445908406220024 | 1.0 | 0.5 | 0.7261308796389632 |
| 676.0 | 0.3632627258496999 | 1.0 | 0.5 | 0.9027963019039356 |
| 677.0 | 0.9360322656138116 | 1.0 | 0.5 | 5.49875294592944 |
| 678.0 | 9.829392285480298e-2 | 1.0 | 0.5 | 0.20693333769178202 |
| 679.0 | 0.16857186470900742 | 1.0 | 0.5 | 0.3692208237524054 |
| 680.0 | 0.6851865756088779 | 1.0 | 0.5 | 2.3115502383155535 |
| 681.0 | 6.0746870332459846e-2 | 1.0 | 0.5 | 0.12534052488294534 |
| 682.0 | 0.13633830378369793 | 1.0 | 0.5 | 0.29314828432216455 |
| 683.0 | 7.392456123744273e-2 | 1.0 | 0.5 | 0.15359916061613507 |
| 684.0 | 0.5086610045902572 | 1.0 | 0.5 | 1.4212419421338762 |
| 685.0 | 6.49975666739836e-2 | 1.0 | 0.5 | 0.13441229441823688 |
| 686.0 | 0.5965150506093662 | 1.0 | 0.5 | 1.8152321842298589 |
| 687.0 | 0.4605739048163946 | 1.0 | 0.5 | 1.2344989825581794 |
| 688.0 | 6.864473076963751e-2 | 1.0 | 0.5 | 0.14222894978542783 |
| 689.0 | 0.8344360964556317 | 1.0 | 0.5 | 3.596796069113207 |
| 690.0 | 8.732363913477414e-2 | 1.0 | 0.5 | 0.18274788003816578 |
| 691.0 | 0.21142843578856352 | 1.0 | 0.5 | 0.4750642335181355 |
| 692.0 | 0.7222864812254323 | 1.0 | 0.5 | 2.56233040927018 |
| 693.0 | 0.8184225707114926 | 1.0 | 0.5 | 3.41214621719645 |
| 694.0 | 0.21034070512424763 | 1.0 | 0.5 | 0.4723073977097867 |
| 695.0 | 0.3162647360277784 | 1.0 | 0.5 | 0.7603689545111227 |
| 696.0 | 0.790802361192664 | 1.0 | 0.5 | 3.1289516672537636 |
| 697.0 | 0.13344125555833664 | 1.0 | 0.5 | 0.28645075407773934 |
| 698.0 | 0.7007503099963927 | 1.0 | 0.5 | 2.4129539409113554 |
| 699.0 | 0.9187168349404677 | 1.0 | 0.5 | 5.019632711418354 |
| 700.0 | 0.4599361344660652 | 1.0 | 0.5 | 1.2321357538196342 |
| 701.0 | 0.27682148640453996 | 1.0 | 0.5 | 0.6481983610575905 |
| 702.0 | 1.800784163218272e-3 | 1.0 | 0.5 | 3.604815048388172e-3 |
| 703.0 | 0.7228478637519237 | 1.0 | 0.5 | 2.566377390466142 |
| 704.0 | 0.2985911645520416 | 1.0 | 0.5 | 0.7093286889612735 |
| 705.0 | 0.17315276773821076 | 1.0 | 0.5 | 0.38027065243919844 |
| 706.0 | 0.14367496112886713 | 1.0 | 0.5 | 0.3102105132154968 |
| 707.0 | 0.8032824900526222 | 1.0 | 0.5 | 3.2519730780123863 |
| 708.0 | 0.9263217228455575 | 1.0 | 0.5 | 5.216094540240102 |
| 709.0 | 0.14390022398042068 | 1.0 | 0.5 | 0.3107366977250743 |
| 710.0 | 0.16040391518009045 | 1.0 | 0.5 | 0.349668708391516 |
| 711.0 | 0.23371462343815053 | 1.0 | 0.5 | 0.5324012487287254 |
| 712.0 | 0.5919728166898646 | 1.0 | 0.5 | 1.7928429620744737 |
| 713.0 | 0.103304290572939 | 1.0 | 0.5 | 0.2180774117665795 |
| 714.0 | 0.9472027518538018 | 1.0 | 0.5 | 5.8825924161427 |
| 715.0 | 2.606352915311938e-2 | 1.0 | 0.5 | 5.2818404940095244e-2 |
| 716.0 | 0.3644003382289155 | 1.0 | 0.5 | 0.9063727529217371 |
| 717.0 | 0.8078186466096073 | 1.0 | 0.5 | 3.298631607703358 |
| 718.0 | 0.945005788294678 | 1.0 | 0.5 | 5.801054682018765 |
| 719.0 | 0.3026751927307447 | 1.0 | 0.5 | 0.7210079384190151 |
| 720.0 | 0.13192709639852174 | 1.0 | 0.5 | 0.28295915504898594 |
| 721.0 | 0.2566043774869836 | 1.0 | 0.5 | 0.5930538192083015 |
| 722.0 | 0.10698594605754641 | 1.0 | 0.5 | 0.22630592066398597 |
| 723.0 | 0.6145249789812167 | 1.0 | 0.5 | 1.9065577687955066 |
| 724.0 | 8.930234303466789e-2 | 1.0 | 0.5 | 0.18708863439196044 |
| 725.0 | 0.5078727259176272 | 1.0 | 0.5 | 1.4180358175685772 |
| 726.0 | 0.6074906579312169 | 1.0 | 0.5 | 1.87038988118759 |
| 727.0 | 0.4328926795524597 | 1.0 | 0.5 | 1.1344134309597795 |
| 728.0 | 0.9199649411490993 | 1.0 | 0.5 | 5.0505810093383765 |
| 729.0 | 0.9554407909442231 | 1.0 | 0.5 | 6.221872867230832 |
| 730.0 | 0.25047456653555367 | 1.0 | 0.5 | 0.5766300562133851 |
| 731.0 | 0.8421247486490246 | 1.0 | 0.5 | 3.6919002124484552 |
| 732.0 | 7.259025420929721e-2 | 1.0 | 0.5 | 0.15071959671184512 |
| 733.0 | 0.11620186500134255 | 1.0 | 0.5 | 0.24705319299034392 |
| 734.0 | 0.30850615207588095 | 1.0 | 0.5 | 0.7378020492564494 |
| 735.0 | 0.1883626739567803 | 1.0 | 0.5 | 0.4174033627979661 |
| 736.0 | 0.6623776199438104 | 1.0 | 0.5 | 2.171654453643791 |
| 737.0 | 0.11290896034135722 | 1.0 | 0.5 | 0.2396153284366351 |
| 738.0 | 0.7422642656298815 | 1.0 | 0.5 | 2.7116410087728458 |
| 739.0 | 0.7587318490143916 | 1.0 | 0.5 | 2.8436926086943526 |
| 740.0 | 0.530336944600492 | 1.0 | 0.5 | 1.5114794895629295 |
| 741.0 | 0.6223687487469611 | 1.0 | 0.5 | 1.9476741705662395 |
| 742.0 | 0.5630377543656311 | 1.0 | 0.5 | 1.6558169640962446 |
| 743.0 | 0.6774789767908294 | 1.0 | 0.5 | 2.2631739132509616 |
| 744.0 | 0.7134140595880138 | 1.0 | 0.5 | 2.499433642548651 |
| 745.0 | 0.10002976757404214 | 1.0 | 0.5 | 0.2107871825741759 |
| 746.0 | 0.567848281252265 | 1.0 | 0.5 | 1.677957103179304 |
| 747.0 | 0.4060863958044679 | 1.0 | 0.5 | 1.0420428353605442 |
| 748.0 | 9.50513656846318e-2 | 1.0 | 0.5 | 0.19975418911112916 |
| 749.0 | 0.16631249236543122 | 1.0 | 0.5 | 0.36379327584656745 |
| 750.0 | 0.8411217147036525 | 1.0 | 0.5 | 3.6792337423769124 |
| 751.0 | 0.8261179212585096 | 1.0 | 0.5 | 3.49875583579596 |
| 752.0 | 2.270068052664842e-2 | 1.0 | 0.5 | 4.5924615942012824e-2 |
| 753.0 | 0.794409914691346 | 1.0 | 0.5 | 3.163741939541678 |
| 754.0 | 9.042111886289728e-2 | 1.0 | 0.5 | 0.18954710912519077 |
| 755.0 | 0.8703489330265705 | 1.0 | 0.5 | 4.0858170747733435 |
| 756.0 | 0.4310407173673936 | 1.0 | 0.5 | 1.1278928138744269 |
| 757.0 | 0.3363067143343986 | 1.0 | 0.5 | 0.819870310803085 |
| 758.0 | 0.5130758701244038 | 1.0 | 0.5 | 1.4392939176921202 |
| 759.0 | 0.7908192098998476 | 1.0 | 0.5 | 3.1291127530653498 |
| 760.0 | 0.13285818054236243 | 1.0 | 0.5 | 0.28510548131242014 |
| 761.0 | 0.20374760571336736 | 1.0 | 0.5 | 0.45567813029380005 |
| 762.0 | 0.5394576445781261 | 1.0 | 0.5 | 1.5507009009747217 |
| 763.0 | 6.022815649686475e-2 | 1.0 | 0.5 | 0.12423630571358278 |
| 764.0 | 6.998298713513784e-2 | 1.0 | 0.5 | 0.1451047991981602 |
| 765.0 | 0.8970007092256013 | 1.0 | 0.5 | 4.546066352923313 |
| 766.0 | 0.4229521552684008 | 1.0 | 0.5 | 1.0996601921991296 |
| 767.0 | 0.8670835830456738 | 1.0 | 0.5 | 4.036069584532981 |
| 768.0 | 0.4225403174878275 | 1.0 | 0.5 | 1.0982333057100466 |
| 769.0 | 0.7514278370027265 | 1.0 | 0.5 | 2.784044162493016 |
| 770.0 | 0.3166625910490667 | 1.0 | 0.5 | 0.7615330624342808 |
| 771.0 | 0.35844011299274825 | 1.0 | 0.5 | 0.8877054892696948 |
| 772.0 | 0.6478614003000068 | 1.0 | 0.5 | 2.08746086347037 |
| 773.0 | 0.7201698321036184 | 1.0 | 0.5 | 2.547144806123709 |
| 774.0 | 0.36834307496040686 | 1.0 | 0.5 | 0.9188177447070995 |
| 775.0 | 0.8093749686530289 | 1.0 | 0.5 | 3.3148939343546955 |
| 776.0 | 0.11152824342387613 | 1.0 | 0.5 | 0.2365048393033555 |
| 777.0 | 0.8664817488175403 | 1.0 | 0.5 | 4.027034195017947 |
| 778.0 | 0.5335069631739435 | 1.0 | 0.5 | 1.525024370448048 |
| 779.0 | 0.3938842132094552 | 1.0 | 0.5 | 1.0013684877054532 |
| 780.0 | 0.10183253557450678 | 1.0 | 0.5 | 0.21479748413816208 |
| 781.0 | 8.053929148343719e-2 | 1.0 | 0.5 | 0.16793593441701318 |
| 782.0 | 0.6987324536890543 | 1.0 | 0.5 | 2.39951310171572 |
| 783.0 | 0.9652504478720477 | 1.0 | 0.5 | 6.719177190917287 |
| 784.0 | 8.352182889441673e-2 | 1.0 | 0.5 | 0.17443405931174336 |
| 785.0 | 0.6848311037778758 | 1.0 | 0.5 | 2.309293210731784 |
| 786.0 | 0.8753831237435886 | 1.0 | 0.5 | 4.165022476660053 |
| 787.0 | 0.5017501580403889 | 1.0 | 0.5 | 1.3933072741604589 |
| 788.0 | 0.5164740417097261 | 1.0 | 0.5 | 1.4533005544713893 |
| 789.0 | 5.34146726412843e-3 | 1.0 | 0.5 | 1.0711567808793232e-2 |
| 790.0 | 0.3318023439651666 | 1.0 | 0.5 | 0.8063425138905206 |
| 791.0 | 0.8980158281288398 | 1.0 | 0.5 | 4.565875310945801 |
| 792.0 | 0.3077912787425361 | 1.0 | 0.5 | 0.7357354970569514 |
| 793.0 | 8.244870505377133e-2 | 1.0 | 0.5 | 0.1720935866356492 |
| 794.0 | 0.9525163323404254 | 1.0 | 0.5 | 6.09473893162687 |
| 795.0 | 0.8150156026591003 | 1.0 | 0.5 | 3.37496759231672 |
| 796.0 | 0.6942073379694466 | 1.0 | 0.5 | 2.3696959635020733 |
| 797.0 | 0.3064154871080699 | 1.0 | 0.5 | 0.7317643647045484 |
| 798.0 | 0.6766043921266462 | 1.0 | 0.5 | 2.257757826055945 |
| 799.0 | 0.8850082431583656 | 1.0 | 0.5 | 4.325789665651913 |
| 800.0 | 0.390824501859813 | 1.0 | 0.5 | 0.9912977570009057 |
| 801.0 | 0.9277232680571199 | 1.0 | 0.5 | 5.254506056261732 |
| 802.0 | 0.9515352841551716 | 1.0 | 0.5 | 6.0538385083107515 |
| 803.0 | 0.931339195694947 | 1.0 | 0.5 | 5.3571535534070165 |
| 804.0 | 0.41693501872311356 | 1.0 | 0.5 | 1.078913277353363 |
| 805.0 | 0.9925072396884476 | 1.0 | 0.5 | 9.787636032854072 |
| 806.0 | 0.6270345104129288 | 1.0 | 0.5 | 1.9725387696659187 |
| 807.0 | 0.9026186077418837 | 1.0 | 0.5 | 4.658240262325699 |
| 808.0 | 0.9572829405210631 | 1.0 | 0.5 | 6.306313838242294 |
| 809.0 | 0.3836448542598979 | 1.0 | 0.5 | 0.9678638925777285 |
| 810.0 | 0.9149835758010022 | 1.0 | 0.5 | 4.929821630573945 |
| 811.0 | 0.9650769640332738 | 1.0 | 0.5 | 6.70921722232391 |
| 812.0 | 0.8333633906893451 | 1.0 | 0.5 | 3.5838796592561692 |
| 813.0 | 0.4659441674707303 | 1.0 | 0.5 | 1.254509780378417 |
| 814.0 | 0.6644565585399186 | 1.0 | 0.5 | 2.1840076964202977 |
| 815.0 | 0.5950530445808047 | 1.0 | 0.5 | 1.8079983894541376 |
| 816.0 | 0.6963681256453612 | 1.0 | 0.5 | 2.383878501957269 |
| 817.0 | 0.7577963228381738 | 1.0 | 0.5 | 2.835952531284403 |
| 818.0 | 5.205609640929987e-2 | 1.0 | 0.5 | 0.10691990381092617 |
| 819.0 | 0.266407912351019 | 1.0 | 0.5 | 0.6196042874911304 |
| 820.0 | 0.2334514658552196 | 1.0 | 0.5 | 0.5317145270071396 |
| 821.0 | 0.3750267027666355 | 1.0 | 0.5 | 0.9400927091701334 |
| 822.0 | 0.6114309357827568 | 1.0 | 0.5 | 1.8905687070019104 |
| 823.0 | 0.15551257370829796 | 1.0 | 0.5 | 0.338050863566619 |
| 824.0 | 0.23813670768097417 | 1.0 | 0.5 | 0.5439762915924459 |
| 825.0 | 0.870266707449991 | 1.0 | 0.5 | 4.084549063402345 |
| 826.0 | 3.154053223157938e-2 | 1.0 | 0.5 | 6.409729506779938e-2 |
| 827.0 | 0.1696763468222191 | 1.0 | 0.5 | 0.3718794212240955 |
| 828.0 | 0.9382537256649148 | 1.0 | 0.5 | 5.5694432798975 |
| 829.0 | 0.5690815690817298 | 1.0 | 0.5 | 1.6836729243990005 |
| 830.0 | 0.40173525398781396 | 1.0 | 0.5 | 1.027443807837522 |
| 831.0 | 0.37414920622824466 | 1.0 | 0.5 | 0.9372865697980035 |
| 832.0 | 0.8470235035676409 | 1.0 | 0.5 | 3.75494197495865 |
| 833.0 | 0.1639959478716596 | 1.0 | 0.5 | 0.35824363773142587 |
| 834.0 | 0.8417875076966232 | 1.0 | 0.5 | 3.6876325230218985 |
| 835.0 | 4.4800017902082656e-2 | 1.0 | 0.5 | 9.166911017102182e-2 |
| 836.0 | 8.371841878440967e-2 | 1.0 | 0.5 | 0.17486311694572496 |
| 837.0 | 0.792681678313694 | 1.0 | 0.5 | 3.1469997619491537 |
| 838.0 | 0.70931724616291 | 1.0 | 0.5 | 2.4710455990059654 |
| 839.0 | 0.9536753385934513 | 1.0 | 0.5 | 6.14416163184868 |
| 840.0 | 0.9602885160673758 | 1.0 | 0.5 | 6.452229730589149 |
| 841.0 | 0.7554874920826832 | 1.0 | 0.5 | 2.8169776284993024 |
| 842.0 | 0.13372214044518893 | 1.0 | 0.5 | 0.28709913578589247 |
| 843.0 | 0.5137668062151739 | 1.0 | 0.5 | 1.442133895114351 |
| 844.0 | 0.7646430480436962 | 1.0 | 0.5 | 2.893303945544842 |
| 845.0 | 0.737797471240314 | 1.0 | 0.5 | 2.677276126613299 |
| 846.0 | 0.2757355366840477 | 1.0 | 0.5 | 0.64519734494416 |
| 847.0 | 0.9070120314664122 | 1.0 | 0.5 | 4.750570329609125 |
| 848.0 | 0.8891902056378072 | 1.0 | 0.5 | 4.399880223610048 |
| 849.0 | 0.6345074545436008 | 1.0 | 0.5 | 2.0130187909368447 |
| 850.0 | 0.7278594124350969 | 1.0 | 0.5 | 2.602872960335896 |
| 851.0 | 0.7746614730325766 | 1.0 | 0.5 | 2.9803028864018035 |
| 852.0 | 0.1615769961852126 | 1.0 | 0.5 | 0.35246505607116085 |
| 853.0 | 0.791392456147658 | 1.0 | 0.5 | 3.134601145736619 |
| 854.0 | 0.6680598448264417 | 1.0 | 0.5 | 2.2056011636299084 |
| 855.0 | 8.13464244932427e-3 | 1.0 | 0.5 | 1.63358183694174e-2 |
| 856.0 | 0.10662505504031572 | 1.0 | 0.5 | 0.2254978301028323 |
| 857.0 | 0.9560295924470773 | 1.0 | 0.5 | 6.248476853892508 |
| 858.0 | 0.852060175832956 | 1.0 | 0.5 | 3.821899362730954 |
| 859.0 | 0.9719898411238275 | 1.0 | 0.5 | 7.150376035205891 |
| 860.0 | 0.2712547072167808 | 1.0 | 0.5 | 0.6328620012818229 |
| 861.0 | 0.5288209915498573 | 1.0 | 0.5 | 1.5050343935333608 |
| 862.0 | 0.7664883160339533 | 1.0 | 0.5 | 2.909046329453154 |
| 863.0 | 0.19497241124386178 | 1.0 | 0.5 | 0.433757460808162 |
| 864.0 | 0.22833492954177637 | 1.0 | 0.5 | 0.5184093393054244 |
| 865.0 | 0.8948424632463157 | 1.0 | 0.5 | 4.504591406380232 |
| 866.0 | 0.8605530887700942 | 1.0 | 0.5 | 3.9401426296279958 |
| 867.0 | 0.15650870586231802 | 1.0 | 0.5 | 0.34041139633305906 |
| 868.0 | 0.45930982890306726 | 1.0 | 0.5 | 1.2298177217350685 |
| 869.0 | 0.3246506528569505 | 1.0 | 0.5 | 0.7850503413401291 |
| 870.0 | 0.9510363593524436 | 1.0 | 0.5 | 6.033354567822288 |
| 871.0 | 0.5038715294705464 | 1.0 | 0.5 | 1.4018407452622141 |
| 872.0 | 5.1203794293428584e-2 | 1.0 | 0.5 | 0.10512249958235921 |
| 873.0 | 0.5544890265721415 | 1.0 | 0.5 | 1.617066801324528 |
| 874.0 | 0.2614676505710716 | 1.0 | 0.5 | 0.6061807474664205 |
| 875.0 | 5.243774821209923e-2 | 1.0 | 0.5 | 0.10772528617673659 |
| 876.0 | 0.7971971719215974 | 1.0 | 0.5 | 3.191042124405132 |
| 877.0 | 0.7919800318693259 | 1.0 | 0.5 | 3.140242406520687 |
| 878.0 | 0.5219283825848018 | 1.0 | 0.5 | 1.4759894609704223 |
| 879.0 | 0.4275949818719734 | 1.0 | 0.5 | 1.1158169290356357 |
| 880.0 | 0.6781991679783914 | 1.0 | 0.5 | 2.267644917806484 |
| 881.0 | 0.8932373960711312 | 1.0 | 0.5 | 4.4742951285035835 |
| 882.0 | 0.5478472624532599 | 1.0 | 0.5 | 1.5874704826077597 |
| 883.0 | 0.5685711063530478 | 1.0 | 0.5 | 1.6813051416100346 |
| 884.0 | 0.9721023498460244 | 1.0 | 0.5 | 7.158425635147378 |
| 885.0 | 0.21573425270663737 | 1.0 | 0.5 | 0.486014705364895 |
| 886.0 | 0.35259191127848954 | 1.0 | 0.5 | 0.8695568868089019 |
| 887.0 | 0.3472690789117947 | 1.0 | 0.5 | 0.853180600685918 |
| 888.0 | 0.5447448280041739 | 1.0 | 0.5 | 1.573794399314694 |
| 889.0 | 0.6727997991506037 | 1.0 | 0.5 | 2.2343661208216705 |
| 890.0 | 0.8696740488652007 | 1.0 | 0.5 | 4.0754333003570125 |
| 891.0 | 0.8734188023436674 | 1.0 | 0.5 | 4.133742596048804 |
| 892.0 | 0.29463161413393557 | 1.0 | 0.5 | 0.6980701589988371 |
| 893.0 | 3.283064353893961e-2 | 1.0 | 0.5 | 6.676332583206068e-2 |
| 894.0 | 0.23328215590600432 | 1.0 | 0.5 | 0.5312728295896356 |
| 895.0 | 0.40670843024467607 | 1.0 | 0.5 | 1.044138629788592 |
| 896.0 | 0.8227097978732368 | 1.0 | 0.5 | 3.45993465796266 |
| 897.0 | 0.7693837137837822 | 1.0 | 0.5 | 2.9340000964909345 |
| 898.0 | 0.5297116607776882 | 1.0 | 0.5 | 1.5088185693552791 |
| 899.0 | 0.7754688592512937 | 1.0 | 0.5 | 2.98748173968772 |
| 900.0 | 0.3248143864258809 | 1.0 | 0.5 | 0.7855352856755076 |
| 901.0 | 0.9401865839181958 | 1.0 | 0.5 | 5.633050587988034 |
| 902.0 | 0.34780122634541877 | 1.0 | 0.5 | 0.8548117918995641 |
| 903.0 | 0.7284097775344618 | 1.0 | 0.5 | 2.6069217674085974 |
| 904.0 | 0.9414922145837621 | 1.0 | 0.5 | 5.677190899121933 |
| 905.0 | 0.9905747699450492 | 1.0 | 0.5 | 9.328730273865226 |
| 906.0 | 0.6248601140341811 | 1.0 | 0.5 | 1.9609125866553052 |
| 907.0 | 2.9604572975377663e-2 | 1.0 | 0.5 | 6.010326765544391e-2 |
| 908.0 | 5.571585670694823e-3 | 1.0 | 0.5 | 1.1174329696467535e-2 |
| 909.0 | 0.6109088431526207 | 1.0 | 0.5 | 1.8878832528951692 |
| 910.0 | 0.4546914044998336 | 1.0 | 0.5 | 1.2128068285824642 |
| 911.0 | 6.989233588498944e-2 | 1.0 | 0.5 | 0.1449098633396298 |
| 912.0 | 7.423451728270214e-2 | 1.0 | 0.5 | 0.1542686696318581 |
| 913.0 | 0.5259803352971397 | 1.0 | 0.5 | 1.4930129428654308 |
| 914.0 | 0.658351708925572 | 1.0 | 0.5 | 2.1479469194904803 |
| 915.0 | 0.7900569250260353 | 1.0 | 0.5 | 3.121837713127503 |
| 916.0 | 0.23734310603381692 | 1.0 | 0.5 | 0.5418940581763712 |
| 917.0 | 0.5669448736548646 | 1.0 | 0.5 | 1.6737804930156697 |
| 918.0 | 0.32179695871984115 | 1.0 | 0.5 | 0.7766171299190336 |
| 919.0 | 0.10866609186678144 | 1.0 | 0.5 | 0.23007233022421117 |
| 920.0 | 0.45705634966886644 | 1.0 | 0.5 | 1.2214994782912616 |
| 921.0 | 0.9167978760940737 | 1.0 | 0.5 | 4.972964807530681 |
| 922.0 | 0.3082714338000597 | 1.0 | 0.5 | 0.7371232913772323 |
| 923.0 | 0.47735097863544884 | 1.0 | 0.5 | 1.2976902549082674 |
| 924.0 | 0.4404786851833078 | 1.0 | 0.5 | 1.161347311537362 |
| 925.0 | 0.7939848260637378 | 1.0 | 0.5 | 3.1596109060410877 |
| 926.0 | 0.2412573034838612 | 1.0 | 0.5 | 0.5521851246673014 |
| 927.0 | 0.3000287671367394 | 1.0 | 0.5 | 0.7134320813856403 |
| 928.0 | 0.6396708208326259 | 1.0 | 0.5 | 2.0414745575053637 |
| 929.0 | 0.7579547542491935 | 1.0 | 0.5 | 2.83726120877795 |
| 930.0 | 0.3396814387067095 | 1.0 | 0.5 | 0.8300657835650016 |
| 931.0 | 0.2892910263130618 | 1.0 | 0.5 | 0.6829845053640449 |
| 932.0 | 0.9011216837211008 | 1.0 | 0.5 | 4.6277306266829745 |
| 933.0 | 0.8210499505161656 | 1.0 | 0.5 | 3.441297130493794 |
| 934.0 | 0.20855830013501853 | 1.0 | 0.5 | 0.4677981203133048 |
| 935.0 | 0.20067661109578638 | 1.0 | 0.5 | 0.4479793460879907 |
| 936.0 | 0.2783975695108384 | 1.0 | 0.5 | 0.6525618840897621 |
| 937.0 | 8.228356433882655e-2 | 1.0 | 0.5 | 0.171733659388771 |
| 938.0 | 0.6509731323562694 | 1.0 | 0.5 | 2.105212750180706 |
| 939.0 | 0.22847862352050963 | 1.0 | 0.5 | 0.5187817997542151 |
| 940.0 | 0.8395983631124934 | 1.0 | 0.5 | 3.6601487571349267 |
| 941.0 | 0.9743748330931115 | 1.0 | 0.5 | 7.328360656210449 |
| 942.0 | 0.9784624250877514 | 1.0 | 0.5 | 7.675912397820349 |
| 943.0 | 0.30092332380443143 | 1.0 | 0.5 | 0.7159896972722992 |
| 944.0 | 0.9500911854862217 | 1.0 | 0.5 | 5.995115296523221 |
| 945.0 | 8.677317380760863e-2 | 1.0 | 0.5 | 0.1815419774519955 |
| 946.0 | 0.7801060191136249 | 1.0 | 0.5 | 3.0292195076905384 |
| 947.0 | 0.8407117298539127 | 1.0 | 0.5 | 3.674079397023548 |
| 948.0 | 2.5783170671583644e-2 | 1.0 | 0.5 | 5.2242765469739556e-2 |
| 949.0 | 0.6060746119884648 | 1.0 | 0.5 | 1.8631875162923288 |
| 950.0 | 0.5958910047385527 | 1.0 | 0.5 | 1.8121412943134505 |
| 951.0 | 0.1746502894487323 | 1.0 | 0.5 | 0.3838961817736315 |
| 952.0 | 0.5840255212924214 | 1.0 | 0.5 | 1.7542627397268897 |
| 953.0 | 0.39662332502701947 | 1.0 | 0.5 | 1.010427217997472 |
| 954.0 | 0.8294609650651269 | 1.0 | 0.5 | 3.5375821291253438 |
| 955.0 | 0.6431589573123884 | 1.0 | 0.5 | 2.060929709875304 |
| 956.0 | 0.985945005032187 | 1.0 | 0.5 | 8.52955486531813 |
| 957.0 | 0.6701686585706427 | 1.0 | 0.5 | 2.218347683497157 |
| 958.0 | 0.9097930895031728 | 1.0 | 0.5 | 4.8112984836026484 |
| 959.0 | 0.9461308232792471 | 1.0 | 0.5 | 5.842393650220309 |
| 960.0 | 5.2337673573452204e-2 | 1.0 | 0.5 | 0.10751407186495446 |
| 961.0 | 0.8460764659780871 | 1.0 | 0.5 | 3.7425986644134888 |
| 962.0 | 0.10514312394105463 | 1.0 | 0.5 | 0.22218297702597922 |
| 963.0 | 0.6975701033741791 | 1.0 | 0.5 | 2.3918115501203125 |
| 964.0 | 0.9576315406611462 | 1.0 | 0.5 | 6.322702154160914 |
| 965.0 | 0.5756726043081537 | 1.0 | 0.5 | 1.714499924122977 |
| 966.0 | 0.14180795869433915 | 1.0 | 0.5 | 0.3058547603470779 |
| 967.0 | 0.476602852426136 | 1.0 | 0.5 | 1.2948294774050015 |
| 968.0 | 0.5735933728030692 | 1.0 | 0.5 | 1.7047237280891907 |
| 969.0 | 0.21074148987963792 | 1.0 | 0.5 | 0.47332273812264397 |
| 970.0 | 0.4040283383810781 | 1.0 | 0.5 | 1.0351243213329395 |
| 971.0 | 0.44809616679520103 | 1.0 | 0.5 | 1.188762926181943 |
| 972.0 | 0.49753260138877997 | 1.0 | 0.5 | 1.376449039078463 |
| 973.0 | 0.3981895324367999 | 1.0 | 0.5 | 1.0156254423601343 |
| 974.0 | 0.4631341447354761 | 1.0 | 0.5 | 1.2440140393123216 |
| 975.0 | 0.7621220718869883 | 1.0 | 0.5 | 2.8719952879613677 |
| 976.0 | 0.35491286990322435 | 1.0 | 0.5 | 0.8767397717764418 |
| 977.0 | 0.5624416059435056 | 1.0 | 0.5 | 1.6530902199238997 |
| 978.0 | 0.9048958001682184 | 1.0 | 0.5 | 4.705564296273839 |
| 979.0 | 0.5871353426885598 | 1.0 | 0.5 | 1.769270891992473 |
| 980.0 | 0.9552974642377905 | 1.0 | 0.5 | 6.215450101021099 |
| 981.0 | 0.11683294268351829 | 1.0 | 0.5 | 0.24848180677799192 |
| 982.0 | 0.804763553103361 | 1.0 | 0.5 | 3.2670878135466657 |
| 983.0 | 0.9017746463190758 | 1.0 | 0.5 | 4.640981825644075 |
| 984.0 | 0.3546593618234335 | 1.0 | 0.5 | 0.8759539607786164 |
| 985.0 | 0.8175329055993971 | 1.0 | 0.5 | 3.4023708537413193 |
| 986.0 | 0.9271651336840998 | 1.0 | 0.5 | 5.239121011041082 |
| 987.0 | 0.43755514801054285 | 1.0 | 0.5 | 1.150924381235035 |
| 988.0 | 0.12335439229044076 | 1.0 | 0.5 | 0.2633049287820745 |
| 989.0 | 0.3823717678509384 | 1.0 | 0.5 | 0.9637371372795477 |
| 990.0 | 0.7134192888551714 | 1.0 | 0.5 | 2.499470136417535 |
| 991.0 | 0.5559806916407091 | 1.0 | 0.5 | 1.6237744603995141 |
| 992.0 | 0.7947950806945909 | 1.0 | 0.5 | 3.167492385499269 |
| 993.0 | 0.8593443260366468 | 1.0 | 0.5 | 3.9228808076956767 |
| 994.0 | 0.9695468897099642 | 1.0 | 0.5 | 6.983134291741286 |
| 995.0 | 9.645307355937993e-2 | 1.0 | 0.5 | 0.20285446358422873 |
| 996.0 | 0.5759218175673997 | 1.0 | 0.5 | 1.7156748964373492 |
| 997.0 | 0.8948271404741518 | 1.0 | 0.5 | 4.504300002523024 |
| 998.0 | 0.8288317043606008 | 1.0 | 0.5 | 3.5302160428926017 |
| 999.0 | 0.7727026977235016 | 1.0 | 0.5 | 2.962992833506503 |
dfExpRand.describe().show() // look sensible
+-------+-----------------+--------------------+----+----+--------------------+
|summary| Id| rand| one|rate| expo_sample|
+-------+-----------------+--------------------+----+----+--------------------+
| count| 1000| 1000|1000|1000| 1000|
| mean| 499.5| 0.5060789944940257| 1.0| 0.5| 2.024647367046763|
| stddev|288.8194360957494| 0.28795893414434537| 0.0| 0.0| 1.9649069025265538|
| min| 0|6.282087728304298E-4| 1.0| 0.5|0.001256812357281...|
| max| 999| 0.9981606891187609| 1.0| 0.5| 12.596728597154938|
+-------+-----------------+--------------------+----+----+--------------------+
val expoSamplesDF = spark.range(1000000000).toDF("Id") // just make a DF of 100 row indices
.select($"Id", rand(seed=1234567) as "rand") // add a column of random numbers in (0,1)
.withColumn("one",lit(1.0))
.withColumn("rate",lit(0.5))
.withColumn("expo_sample", -($"one" / $"rate") * log($"one" - $"rand"))
expoSamplesDF: org.apache.spark.sql.DataFrame = [Id: bigint, rand: double ... 3 more fields]
expoSamplesDF.describe().show()
+-------+--------------------+--------------------+----------+----------+--------------------+
|summary| Id| rand| one| rate| expo_sample|
+-------+--------------------+--------------------+----------+----------+--------------------+
| count| 1000000000| 1000000000|1000000000|1000000000| 1000000000|
| mean|4.9999999907595944E8| 0.49999521358240573| 1.0| 0.5| 1.9999814119383708|
| stddev|2.8867513473915565E8| 0.2886816216562552| 0.0| 0.0| 2.0000011916404206|
| min| 0|1.584746778249268...| 1.0| 0.5|3.169493556749679...|
| max| 999999999| 0.9999999996809935| 1.0| 0.5| 43.731619350261035|
+-------+--------------------+--------------------+----------+----------+--------------------+
Using sql.functions and sql.expr for expressions, one can write a much simpler syntax to get Exponetially distributed samples.
See: - spark/sql/expressions/ docs - https://sparkbyexamples.com/spark/spark-sql-functions/
import org.apache.spark.sql.functions._
import org.apache.spark.sql.functions.expr
val expoSamplesDFAsSQLExp = spark.range(1000000000).toDF("Id") // just make a DF of 100 row indices
.select($"Id", rand(seed=1234567) as "random") // add a column of random numbers in (0,1)
.withColumn("expo_sample", expr("-(1.0 / 0.5) * log(1.0 - random)"))
import org.apache.spark.sql.functions._
import org.apache.spark.sql.functions.expr
expoSamplesDFAsSQLExp: org.apache.spark.sql.DataFrame = [Id: bigint, random: double ... 1 more field]
expoSamplesDFAsSQLExp.describe().show()
+-------+--------------------+--------------------+--------------------+
|summary| Id| random| expo_sample|
+-------+--------------------+--------------------+--------------------+
| count| 1000000000| 1000000000| 1000000000|
| mean|4.9999999907595944E8| 0.49999521358240573| 1.9999814119383708|
| stddev|2.8867513473915565E8| 0.2886816216562552| 2.0000011916404206|
| min| 0|1.584746778249268...|3.169493556749679...|
| max| 999999999| 0.9999999996809935| 43.731619350261035|
+-------+--------------------+--------------------+--------------------+
Approximating Pi with Monte Carlo simulations
Uisng RDDs directly, let's estimate Pi.
//Calculate pi with Monte Carlo estimation
import scala.math.random
//make a very large unique set of 1 -> n
val partitions = 2
val n = math.min(100000L * partitions, Int.MaxValue).toInt
val xs = 1 until n
//split up n into the number of partitions we can use
val rdd = sc.parallelize(xs, partitions).setName("'N values rdd'")
//generate a random set of points within a 2x2 square
val sample = rdd.map { i =>
val x = random * 2 - 1
val y = random * 2 - 1
(x, y)
}.setName("'Random points rdd'")
//points w/in the square also w/in the center circle of r=1
val inside = sample.filter { case (x, y) => (x * x + y * y < 1) }.setName("'Random points inside circle'")
val count = inside.count()
//Area(circle)/Area(square) = inside/n => pi=4*inside/n
println("Pi is roughly " + 4.0 * count / n)
Pi is roughly 3.1387
import scala.math.random
partitions: Int = 2
n: Int = 200000
xs: scala.collection.immutable.Range = Range 1 until 200000
rdd: org.apache.spark.rdd.RDD[Int] = 'N values rdd' ParallelCollectionRDD[192] at parallelize at command-2971213210276568:10
sample: org.apache.spark.rdd.RDD[(Double, Double)] = 'Random points rdd' MapPartitionsRDD[193] at map at command-2971213210276568:13
inside: org.apache.spark.rdd.RDD[(Double, Double)] = 'Random points inside circle' MapPartitionsRDD[194] at filter at command-2971213210276568:20
count: Long = 156935
Doing it in PySpark is just as easy. This may be needed if there are pyhton libraries you want to take advantage of in Spark.
# # Estimating $\pi$
#
# This PySpark example shows you how to estimate $\pi$ in parallel
# using Monte Carlo integration.
from __future__ import print_function
import sys
from random import random
from operator import add
partitions = 2
n = 100000 * partitions
def f(_):
x = random() * 2 - 1
y = random() * 2 - 1
return 1 if x ** 2 + y ** 2 < 1 else 0
# To access the associated SparkContext
count = spark.sparkContext.parallelize(range(1, n + 1), partitions).map(f).reduce(add)
print("Pi is roughly %f" % (4.0 * count / n))
The following is from this google turotial.
Programming a Monte Carlo simulation in Scala
Monte Carlo, of course, is famous as a gambling destination. In this section, you use Scala to create a simulation that models the mathematical advantage that a casino enjoys in a game of chance. The "house edge" at a real casino varies widely from game to game; it can be over 20% in keno, for example. This tutorial creates a simple game where the house has only a one-percent advantage. Here's how the game works:
- The player places a bet, consisting of a number of chips from a bankroll fund.
- The player rolls a 100-sided die (how cool would that be?).
- If the result of the roll is a number from 1 to 49, the player wins.
- For results 50 to 100, the player loses the bet.
You can see that this game creates a one-percent disadvantage for the player: in 51 of the 100 possible outcomes for each roll, the player loses.
Follow these steps to create and run the game:
val STARTING_FUND = 10
val STAKE = 1 // the amount of the bet
val NUMBER_OF_GAMES = 25
def rollDie: Int = {
val r = scala.util.Random
r.nextInt(99) + 1
}
def playGame(stake: Int): (Int) = {
val faceValue = rollDie
if (faceValue < 50)
(2*stake)
else
(0)
}
// Function to play the game multiple times
// Returns the final fund amount
def playSession(
startingFund: Int = STARTING_FUND,
stake: Int = STAKE,
numberOfGames: Int = NUMBER_OF_GAMES):
(Int) = {
// Initialize values
var (currentFund, currentStake, currentGame) = (startingFund, 0, 1)
// Keep playing until number of games is reached or funds run out
while (currentGame <= numberOfGames && currentFund > 0) {
// Set the current bet and deduct it from the fund
currentStake = math.min(stake, currentFund)
currentFund -= currentStake
// Play the game
val (winnings) = playGame(currentStake)
// Add any winnings
currentFund += winnings
// Increment the loop counter
currentGame += 1
}
(currentFund)
}
STARTING_FUND: Int = 10
STAKE: Int = 1
NUMBER_OF_GAMES: Int = 25
rollDie: Int
playGame: (stake: Int)Int
playSession: (startingFund: Int, stake: Int, numberOfGames: Int)Int
Enter the following code to play the game 25 times, which is the default value for NUMBER_OF_GAMES.
playSession()
res31: Int = 5
Your bankroll started with a value of 10 units. Is it higher or lower, now?
Now simulate 10,000 players betting 100 chips per game. Play 10,000 games in a session. This Monte Carlo simulation calculates the probability of losing all your money before the end of the session. Enter the follow code:
(sc.parallelize(1 to 10000, 500)
.map(i => playSession(100000, 100, 250000))
.map(i => if (i == 0) 1 else 0)
.reduce(_+_)/10000.0)
res32: Double = 0.9992
Note that the syntax .reduce(_+_) is shorthand in Scala for aggregating by using a summing function.
The preceding code performs the following steps:
- Creates an RDD with the results of playing the session.
- Replaces bankrupt players' results with the number 1 and nonzero results with the number 0.
- Sums the count of bankrupt players.
- Divides the count by the number of players.
A typical result might be:
res32: Double = 0.9992
Which represents a near guarantee of losing all your money, even though the casino had only a one-percent advantage.
Project Ideas
Try to create a scalable simulation of interest to you.
Here are some mature projects:
- https://github.com/zishanfu/GeoSparkSim
- https://github.com/srbaird/mc-var-spark
See a more complete VaR modeling: - https://databricks.com/blog/2020/05/27/modernizing-risk-management-part-1-streaming-data-ingestion-rapid-model-development-and-monte-carlo-simulations-at-scale.html