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Posted to commits@commons.apache.org by ah...@apache.org on 2022/01/23 08:30:07 UTC
[commons-statistics] branch master updated (876b5ec -> 955b664)
This is an automated email from the ASF dual-hosted git repository.
aherbert pushed a change to branch master
in repository https://gitbox.apache.org/repos/asf/commons-statistics.git.
from 876b5ec STATISTICS-25: T-dist to use the complement of the regularized beta
new a329006 Use ParameterizedTest
new 955b664 Beta distribution to use beta functions from Commons Numbers
The 2 revisions listed above as "new" are entirely new to this
repository and will be described in separate emails. The revisions
listed as "add" were already present in the repository and have only
been added to this reference.
Summary of changes:
.../statistics/distribution/BetaDistribution.java | 19 ++++++++++--------
.../distribution/BetaDistributionTest.java | 23 +++++++++++-----------
.../statistics/distribution/test.beta.1.properties | 10 +++++-----
.../distribution/test.beta.10.properties | 12 +++++------
.../distribution/test.beta.11.properties | 12 +++++------
.../distribution/test.beta.12.properties | 4 ++--
.../distribution/test.beta.13.properties | 6 +++---
.../distribution/test.beta.14.properties | 4 ++--
.../distribution/test.beta.15.properties | 6 +++---
.../distribution/test.beta.16.properties | 4 ++--
.../distribution/test.beta.17.properties | 4 ++--
.../distribution/test.beta.18.properties | 4 ++--
.../distribution/test.beta.19.properties | 6 +++---
.../statistics/distribution/test.beta.2.properties | 12 +++++------
.../distribution/test.beta.20.properties | 4 ++--
.../distribution/test.beta.21.properties | 4 ++--
.../distribution/test.beta.22.properties | 4 ++--
.../distribution/test.beta.23.properties | 6 +++---
.../distribution/test.beta.24.properties | 4 ++--
.../distribution/test.beta.25.properties | 4 ++--
.../statistics/distribution/test.beta.3.properties | 12 +++++------
.../statistics/distribution/test.beta.4.properties | 12 +++++------
.../statistics/distribution/test.beta.5.properties | 12 +++++------
.../statistics/distribution/test.beta.6.properties | 12 +++++------
.../statistics/distribution/test.beta.7.properties | 8 ++++----
.../statistics/distribution/test.beta.8.properties | 12 +++++------
.../statistics/distribution/test.beta.9.properties | 12 +++++------
27 files changed, 118 insertions(+), 114 deletions(-)
[commons-statistics] 01/02: Use ParameterizedTest
Posted by ah...@apache.org.
This is an automated email from the ASF dual-hosted git repository.
aherbert pushed a commit to branch master
in repository https://gitbox.apache.org/repos/asf/commons-statistics.git
commit a329006d473881055eb8f1ee51f7d77c555f72ba
Author: Alex Herbert <ah...@apache.org>
AuthorDate: Sun Jan 23 07:14:54 2022 +0000
Use ParameterizedTest
---
.../distribution/BetaDistributionTest.java | 23 +++++++++++-----------
1 file changed, 12 insertions(+), 11 deletions(-)
diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/BetaDistributionTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/BetaDistributionTest.java
index f18c632..d6522de 100644
--- a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/BetaDistributionTest.java
+++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/BetaDistributionTest.java
@@ -24,6 +24,8 @@ import org.apache.commons.math3.stat.StatUtils;
import org.apache.commons.math3.stat.inference.GTest;
import org.junit.jupiter.api.Assertions;
import org.junit.jupiter.api.Test;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.CsvSource;
/**
* Test cases for {@link BetaDistribution}.
@@ -47,7 +49,7 @@ class BetaDistributionTest extends BaseContinuousDistributionTest {
@Override
protected double getRelativeTolerance() {
- return 5e-14;
+ return 1e-14;
}
@Override
@@ -115,16 +117,15 @@ class BetaDistributionTest extends BaseContinuousDistributionTest {
() -> "survival function not precise at " + value + " for a=" + alpha + " & b=" + beta);
}
- @Test
- void testLogDensityPrecondition1() {
- final BetaDistribution dist = BetaDistribution.of(0.5, 3);
- Assertions.assertEquals(Double.POSITIVE_INFINITY, dist.logDensity(0.0));
- }
-
- @Test
- void testLogDensityPrecondition2() {
- final BetaDistribution dist = BetaDistribution.of(2, 0.5);
- Assertions.assertEquals(Double.POSITIVE_INFINITY, dist.logDensity(1.0));
+ @ParameterizedTest
+ @CsvSource({
+ "0.5, 3, 0, Infinity",
+ "2, 0.5, 1, Infinity",
+ })
+ void testLogDensityPrecondition(double a, double b, double x, double expected) {
+ final BetaDistribution dist = BetaDistribution.of(a, b);
+ Assertions.assertEquals(expected, dist.density(x));
+ Assertions.assertEquals(Math.log(expected), dist.logDensity(x));
}
@Test
[commons-statistics] 02/02: Beta distribution to use beta functions from Commons Numbers
Posted by ah...@apache.org.
This is an automated email from the ASF dual-hosted git repository.
aherbert pushed a commit to branch master
in repository https://gitbox.apache.org/repos/asf/commons-statistics.git
commit 955b6640ce8076d1fc474295c0607ddb38525155
Author: Alex Herbert <ah...@apache.org>
AuthorDate: Sun Jan 23 08:28:31 2022 +0000
Beta distribution to use beta functions from Commons Numbers
Use RegularizedBeta.derivative.
Use LogBeta in place of 3 LogGamma calls.
Test data has been updated to 17 significant figures. Test tolerance
lowered from 5e-14 to 1e-14.
---
.../statistics/distribution/BetaDistribution.java | 19 +++++++++++--------
.../statistics/distribution/test.beta.1.properties | 10 +++++-----
.../statistics/distribution/test.beta.10.properties | 12 ++++++------
.../statistics/distribution/test.beta.11.properties | 12 ++++++------
.../statistics/distribution/test.beta.12.properties | 4 ++--
.../statistics/distribution/test.beta.13.properties | 6 +++---
.../statistics/distribution/test.beta.14.properties | 4 ++--
.../statistics/distribution/test.beta.15.properties | 6 +++---
.../statistics/distribution/test.beta.16.properties | 4 ++--
.../statistics/distribution/test.beta.17.properties | 4 ++--
.../statistics/distribution/test.beta.18.properties | 4 ++--
.../statistics/distribution/test.beta.19.properties | 6 +++---
.../statistics/distribution/test.beta.2.properties | 12 ++++++------
.../statistics/distribution/test.beta.20.properties | 4 ++--
.../statistics/distribution/test.beta.21.properties | 4 ++--
.../statistics/distribution/test.beta.22.properties | 4 ++--
.../statistics/distribution/test.beta.23.properties | 6 +++---
.../statistics/distribution/test.beta.24.properties | 4 ++--
.../statistics/distribution/test.beta.25.properties | 4 ++--
.../statistics/distribution/test.beta.3.properties | 12 ++++++------
.../statistics/distribution/test.beta.4.properties | 12 ++++++------
.../statistics/distribution/test.beta.5.properties | 12 ++++++------
.../statistics/distribution/test.beta.6.properties | 12 ++++++------
.../statistics/distribution/test.beta.7.properties | 8 ++++----
.../statistics/distribution/test.beta.8.properties | 12 ++++++------
.../statistics/distribution/test.beta.9.properties | 12 ++++++------
26 files changed, 106 insertions(+), 103 deletions(-)
diff --git a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/BetaDistribution.java b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/BetaDistribution.java
index 7587966..e2b1012 100644
--- a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/BetaDistribution.java
+++ b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/BetaDistribution.java
@@ -17,7 +17,7 @@
package org.apache.commons.statistics.distribution;
import org.apache.commons.numbers.gamma.RegularizedBeta;
-import org.apache.commons.numbers.gamma.LogGamma;
+import org.apache.commons.numbers.gamma.LogBeta;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.rng.sampling.distribution.ChengBetaSampler;
@@ -46,8 +46,8 @@ public final class BetaDistribution extends AbstractContinuousDistribution {
private final double alpha;
/** Second shape parameter. */
private final double beta;
- /** Normalizing factor used in density computations.*/
- private final double z;
+ /** Normalizing factor used in log density computations. log(beta(a, b)). */
+ private final double logBeta;
/** Cached value for inverse probability function. */
private final double mean;
/** Cached value for inverse probability function. */
@@ -61,7 +61,7 @@ public final class BetaDistribution extends AbstractContinuousDistribution {
double beta) {
this.alpha = alpha;
this.beta = beta;
- z = LogGamma.value(alpha) + LogGamma.value(beta) - LogGamma.value(alpha + beta);
+ logBeta = LogBeta.value(alpha, beta);
final double alphabetasum = alpha + beta;
mean = alpha / alphabetasum;
variance = (alpha * beta) / ((alphabetasum * alphabetasum) * (alphabetasum + 1));
@@ -111,7 +111,10 @@ public final class BetaDistribution extends AbstractContinuousDistribution {
*/
@Override
public double density(double x) {
- return Math.exp(logDensity(x));
+ if (x < 0 || x > 1) {
+ return 0;
+ }
+ return RegularizedBeta.derivative(x, alpha, beta);
}
/** {@inheritDoc}
@@ -133,7 +136,7 @@ public final class BetaDistribution extends AbstractContinuousDistribution {
}
// Special case of cancellation: x^(a-1) (1-x)^(b-1) / B(a, b)
if (alpha == 1) {
- return -z;
+ return -logBeta;
}
return Double.NEGATIVE_INFINITY;
} else if (x == 1) {
@@ -145,13 +148,13 @@ public final class BetaDistribution extends AbstractContinuousDistribution {
}
// Special case of cancellation: x^(a-1) (1-x)^(b-1) / B(a, b)
if (beta == 1) {
- return -z;
+ return -logBeta;
}
return Double.NEGATIVE_INFINITY;
} else {
final double logX = Math.log(x);
final double log1mX = Math.log1p(-x);
- return (alpha - 1) * logX + (beta - 1) * log1mX - z;
+ return (alpha - 1) * logX + (beta - 1) * log1mX - logBeta;
}
}
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.1.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.1.properties
index f034f92..f3be562 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.1.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.1.properties
@@ -16,10 +16,10 @@
parameters = 0.1 0.1
# Computed using Matlab
mean = 0.5
-variance = 0.208333333333333
+variance = 0.20833333333333334
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.406385093936276 0.439709190223346 0.462804186115552 0.482120045609328 0.500000000000000 0.517879954390672 0.537195813884448 0.560290809776654 0.593614906063724 1 1
-pdf.values = 0 Infinity 0.442988958666524 0.263938762316795 0.206639715373253 0.183240320487425 0.176630277977874 0.183240320487425 0.206639715373253 0.263938762316795 0.442988958666524 Infinity 0
-sf.values = 1 1 0.593614906063724 0.560290809776654 0.537195813884448 0.517879954390672 0.500000000000000 0.482120045609328 0.462804186115552 0.439709190223346 0.406385093936276 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.40638509393627625 0.43970919022334581 0.46280418611555235 0.48212004560932825 0.5 0.5178799543906718 0.5371958138844477 0.56029080977665435 0.59361490606372369 1 1
+pdf.values = 0 Inf 0.44298895866652355 0.26393876231679497 0.20663971537325304 0.18324032048742506 0.17663027797787384 0.18324032048742506 0.20663971537325304 0.26393876231679503 0.44298895866652377 Inf 0
+sf.values = 1 1 0.59361490606372369 0.56029080977665413 0.5371958138844477 0.5178799543906718 0.49999999999999994 0.48212004560932825 0.46280418611555235 0.4397091902233457 0.40638509393627625 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.10.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.10.properties
index fbe8b60..f2ffbf8 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.10.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.10.properties
@@ -15,11 +15,11 @@
parameters = 0.5 4.0
# Computed using Matlab
-mean = 0.111111111111111
-variance = 0.017957351290685
+mean = 0.1111111111111111
+variance = 0.017957351290684626
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.626625082597741 0.804984471899925 0.898778484206134 0.950264436880598 0.977796095859523 0.991483736629099 0.997455625383593 0.999522385942406 0.999971488851370 1 1
-pdf.values = 0 Inf 2.52142232809988 1.25219806739988 0.684938469046825 0.373544048607390 0.193349510480697 0.0903696114115064 0.0352965948694063 0.00978279740156159 0.00115291373026972 0 0
-sf.values = 1 1 0.373374917402260 0.195015528100075 0.101221515793866 0.0497355631194020 0.0222039041404773 0.00851626337090129 0.00254437461640681 0.000477614057594006 2.85111486302981e-05 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.62662508259774052 0.80498447189992484 0.89877848420613371 0.95026443688059803 0.97779609585952276 0.99148373662909872 0.99745562538359323 0.99952238594240594 0.99997148885136966 1 1
+pdf.values = 0 Inf 2.5214223280998818 1.2521980673998827 0.68493846904682498 0.37354404860738988 0.19334951048069668 0.090369611411506442 0.035296594869406332 0.0097827974015615871 0.0011529137302697221 0 0
+sf.values = 1 1 0.37337491740225948 0.19501552810007516 0.10122151579386635 0.049735563119401985 0.022203904140477276 0.0085162633709012919 0.0025443746164068113 0.00047761405759400575 2.8511148630298118e-05 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.11.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.11.properties
index 2231ec7..6af1d17 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.11.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.11.properties
@@ -15,11 +15,11 @@
parameters = 1.0 0.1
# Computed using Matlab
-mean = 0.909090909090909
-variance = 0.039354584809130
+mean = 0.90909090909090906
+variance = 0.039354584809130261
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.0104807417937856 0.0220672314570715 0.0350389048801824 0.0497997834943236 0.0669670084631926 0.0875564634445192 0.113431849434787 0.148660077479215 0.205671765275719 1 1
-pdf.values = 0 0.100000000000000 0.109946584245135 0.122241596067866 0.137851585017117 0.158366702750946 0.186606598307362 0.228110884138870 0.295522716855071 0.425669961260392 0.794328234724282 Inf 0
-sf.values = 1 1 0.989519258206214 0.977932768542929 0.964961095119818 0.950200216505676 0.933032991536807 0.912443536555481 0.886568150565213 0.851339922520785 0.794328234724282 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.010480741793785622 0.022067231457071509 0.035038904880182388 0.049799783494323582 0.066967008463192548 0.087556463444519186 0.11343184943478668 0.14866007747921539 0.20567176527571851 1 1
+pdf.values = 0 0.10000000000000001 0.10994658424513494 0.1222415960678661 0.13785158501711683 0.15836670275094611 0.18660659830736154 0.22811088413887023 0.2955227168550712 0.42566996126039247 0.79432823472428193 Inf 0
+sf.values = 1 1 0.98951925820621434 0.97793276854292854 0.96496109511981765 0.95020021650567643 0.93303299153680741 0.91244353655548083 0.88656815056521332 0.85133992252078461 0.79432823472428149 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.12.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.12.properties
index 7abc208..8b5a9a4 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.12.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.12.properties
@@ -19,7 +19,7 @@ mean = 0.66666666666666663
variance = 0.088888888888888892
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.051316701949486218 0.10557280900008409 0.16333997346592444 0.22540333075851654 0.2928932188134527 0.36754446796632401 0.45227744249483381 0.55278640450004213 0.68377223398316211 1 1
pdf.values = 0 0.5 0.52704627669472992 0.55901699437494745 0.59761430466719689 0.64549722436790291 0.70710678118654768 0.79056941504209488 0.9128709291752769 1.1180339887498951 1.58113883008419 Inf 0
-sf.values = 1 1 0.948683298050514 0.894427190999916 0.836660026534076 0.774596669241483 0.707106781186547 0.632455532033676 0.547722557505166 0.447213595499958 0.316227766016838 0 0
+sf.values = 1 1 0.94868329805051377 0.89442719099991597 0.83666002653407556 0.7745966692414834 0.70710678118654724 0.63245553203367599 0.54772255750516619 0.44721359549995793 0.31622776601683789 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.13.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.13.properties
index d867bca..643df3b 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.13.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.13.properties
@@ -19,7 +19,7 @@ mean = 0.5
variance = 0.083333333333333329
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0.0, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.10000000000000003 0.20000000000000004 0.29999999999999993 0.40000000000000008 0.5 0.59999999999999987 0.69999999999999996 0.80000000000000004 0.90000000000000002 1 1
pdf.values = 0 1 1 1 1 1 1 1 1 1 1 1 0
-sf.values = 1 1 0.900000000000000 0.800000000000000 0.700000000000000 0.600000000000000 0.500000000000000 0.400000000000000 0.300000000000000 0.200000000000000 0.100000000000000 0 0
+sf.values = 1 1 0.89999999999999991 0.79999999999999993 0.70000000000000007 0.59999999999999987 0.5 0.40000000000000008 0.30000000000000004 0.19999999999999996 0.099999999999999964 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.14.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.14.properties
index e238b8b..9f86afd 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.14.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.14.properties
@@ -19,7 +19,7 @@ mean = 0.33333333333333331
variance = 0.055555555555555552
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.19000000000000006 0.35999999999999999 0.51000000000000023 0.64000000000000001 0.75 0.83999999999999997 0.90999999999999992 0.95999999999999996 0.98999999999999999 1 1
pdf.values = 0 2 1.8 1.6000000000000001 1.3999999999999999 1.2 1 0.80000000000000004 0.60000000000000009 0.39999999999999991 0.19999999999999996 0 0
-sf.values = 1 1 0.810000000000000 0.640000000000000 0.490000000000000 0.360000000000000 0.250000000000000 0.160000000000000 0.0900000000000000 0.0400000000000000 0.0100000000000000 0 0
+sf.values = 1 1 0.80999999999999994 0.64000000000000001 0.48999999999999977 0.35999999999999999 0.25000000000000006 0.16000000000000003 0.090000000000000024 0.039999999999999994 0.01 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.15.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.15.properties
index 5dc0e16..ec23362 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.15.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.15.properties
@@ -16,10 +16,10 @@
parameters = 1.0 4.0
# Computed using Matlab
mean = 0.20000000000000001
-variance = 0.026666666666666672
+variance = 0.026666666666666668
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.34389999999999987 0.59040000000000026 0.75990000000000002 0.87040000000000006 0.9375 0.97440000000000004 0.9919 0.99839999999999995 0.99990000000000001 1 1
pdf.values = 0 4 2.9160000000000008 2.0480000000000005 1.3719999999999999 0.8640000000000001 0.50000000000000022 0.25600000000000012 0.10800000000000005 0.031999999999999994 0.0039999999999999983 0 0
-sf.values = 1 1 0.656100000000000 0.409600000000000 0.240100000000000 0.129600000000000 0.0625000000000000 0.0256000000000000 0.00810000000000000 0.00160000000000000 9.99999999999999e-05 0 0
+sf.values = 1 1 0.65610000000000013 0.40959999999999974 0.24009999999999992 0.12959999999999997 0.062500000000000028 0.025600000000000001 0.008100000000000003 0.0016000000000000007 9.9999999999999869e-05 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.16.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.16.properties
index 1eaf518..ffa028e 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.16.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.16.properties
@@ -19,7 +19,7 @@ mean = 0.95238095238095233
variance = 0.01462950771706532
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.00058554921172346416 0.0025085760862129224 0.0060900720265878658 0.011791774834096561 0.020315358886352223 0.032809851251190306 0.051372078895221711 0.080552883677552423 0.13418222415053294 1 1
pdf.values = 0 0 0.012094124266964851 0.026893151134930546 0.045491023055648561 0.069681349210416327 0.10263362906904888 0.15055318353165442 0.22755249197840488 0.37458956590914555 0.78638495237703931 Inf 0
-sf.values = 1 1 0.999414450788277 0.997491423913787 0.993909927973412 0.988208225165904 0.979684641113648 0.967190148748810 0.948627921104778 0.919447116322448 0.865817775849467 0 0
+sf.values = 1 1 0.99941445078827651 0.99749142391378709 0.99390992797341216 0.98820822516590345 0.97968464111364773 0.96719014874880971 0.94862792110477834 0.91944711632244758 0.86581777584946706 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.17.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.17.properties
index bca61be..66fec5e 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.17.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.17.properties
@@ -19,7 +19,7 @@ mean = 0.80000000000000004
variance = 0.045714285714285714
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.0038825370469605129 0.016130089900092535 0.0378409694858131 0.070483996910219962 0.1161165235168156 0.17780780835622126 0.26057454736802566 0.37390096630005898 0.54146973927558506 1 1
pdf.values = 0 0 0.079056941504209499 0.16770509831248426 0.26892643710023856 0.38729833462074181 0.53033008588991071 0.71151247353788538 0.95851447563404069 1.3416407864998741 2.1345374206136567 Inf 0
-sf.values = 1 1 0.996117462953040 0.983869910099907 0.962159030514187 0.929516003089780 0.883883476483184 0.822192191643779 0.739425452631974 0.626099033699941 0.458530260724415 0 0
+sf.values = 1 1 0.99611746295303949 0.98386991009990743 0.96215903051418694 0.92951600308978 0.88388347648318444 0.82219219164377877 0.73942545263197434 0.62609903369994102 0.45853026072441494 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.18.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.18.properties
index 2a56be7..e56f538 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.18.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.18.properties
@@ -19,7 +19,7 @@ mean = 0.66666666666666663
variance = 0.055555555555555552
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.010000000000000009 0.040000000000000029 0.089999999999999983 0.16000000000000003 0.25000000000000006 0.35999999999999999 0.48999999999999977 0.64000000000000012 0.81000000000000005 1 1
pdf.values = 0 0 0.20000000000000007 0.40000000000000002 0.59999999999999987 0.80000000000000004 1 1.2 1.3999999999999999 1.6000000000000001 1.8 2 0
-sf.values = 1 1 0.990000000000000 0.960000000000000 0.910000000000000 0.840000000000000 0.750000000000000 0.640000000000000 0.510000000000000 0.360000000000000 0.190000000000000 0 0
+sf.values = 1 1 0.98999999999999999 0.95999999999999996 0.91000000000000003 0.83999999999999997 0.75 0.64000000000000001 0.51000000000000023 0.35999999999999988 0.18999999999999995 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.19.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.19.properties
index e4c3a7b..c1a871b 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.19.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.19.properties
@@ -16,10 +16,10 @@
parameters = 2.0 2.0
# Computed using Matlab
mean = 0.5
-variance = 0.049999999999999996
+variance = 0.050000000000000003
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.028000000000000008 0.10400000000000008 0.2159999999999998 0.35200000000000004 0.5 0.64799999999999991 0.78400000000000003 0.89600000000000002 0.97199999999999998 1 1
pdf.values = 0 0 0.54000000000000004 0.96000000000000008 1.2599999999999998 1.4400000000000002 1.5 1.4400000000000002 1.26 0.95999999999999985 0.53999999999999981 0 0
-sf.values = 1 1 0.972000000000000 0.896000000000000 0.784000000000000 0.648000000000000 0.500000000000000 0.352000000000000 0.216000000000000 0.104000000000000 0.0280000000000000 0 0
+sf.values = 1 1 0.97199999999999998 0.89599999999999991 0.78400000000000025 0.64799999999999991 0.5 0.35200000000000004 0.21599999999999997 0.10399999999999997 0.027999999999999983 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.2.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.2.properties
index 66edffa..ad58eb6 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.2.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.2.properties
@@ -15,11 +15,11 @@
parameters = 0.1 0.5
# Computed using Matlab
-mean = 0.166666666666667
-variance = 0.086805555555556
+mean = 0.16666666666666669
+variance = 0.086805555555555566
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.704833622051221 0.759304219445319 0.795176530367368 0.823494838536657 0.848001712399770 0.870603436980005 0.892658587836406 0.915640640375004 0.942366288315790 1 1
-pdf.values = 0 Infinity 0.739458664410644 0.420303655461325 0.311944266317603 0.260079151351645 0.233065049073820 0.221140881157234 0.222272786172717 0.241401058813832 0.307055512274369 Infinity 0
-sf.values = 1 1 0.295166377948779 0.240695780554681 0.204823469632632 0.176505161463343 0.151998287600230 0.129396563019995 0.107341412163595 0.0843593596249964 0.0576337116842096 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.70483362205122113 0.75930421944531856 0.79517653036736813 0.82349483853665706 0.84800171239977029 0.87060343698000509 0.8926585878364055 0.91564064037500359 0.94236628831579039 1 1
+pdf.values = 0 Inf 0.73945866441064423 0.42030365546132503 0.31194426631760319 0.26007915135164533 0.23306504907381986 0.22114088115723446 0.22227278617271659 0.24140105881383236 0.30705551227436928 Inf 0
+sf.values = 1 1 0.29516637794877887 0.24069578055468144 0.20482346963263187 0.17650516146334294 0.15199828760022974 0.12939656301999489 0.10734141216359451 0.084359359624996383 0.057633711684209582 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.20.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.20.properties
index b2c9d57..8694520 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.20.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.20.properties
@@ -19,7 +19,7 @@ mean = 0.33333333333333331
variance = 0.031746031746031744
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.081459999999999991 0.26272000000000001 0.47177999999999987 0.6630400000000003 0.81250000000000011 0.91295999999999999 0.96921999999999997 0.99328000000000005 0.99953999999999998 1 1
pdf.values = 0 0 1.458 2.0479999999999996 2.0579999999999994 1.7279999999999998 1.25 0.7679999999999999 0.378 0.12799999999999995 0.017999999999999985 0 0
-sf.values = 1 1 0.918540000000000 0.737280000000000 0.528220000000000 0.336960000000000 0.187500000000000 0.0870400000000000 0.0307800000000000 0.00671999999999999 0.000460000000000000 0 0
+sf.values = 1 1 0.91854000000000002 0.73727999999999994 0.52822000000000013 0.33695999999999976 0.18749999999999992 0.087040000000000006 0.030780000000000002 0.0067199999999999925 0.00045999999999999969 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.21.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.21.properties
index 9d87d65..9f80546 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.21.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.21.properties
@@ -19,7 +19,7 @@ mean = 0.97560975609756106
variance = 0.0046657568440820708
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 3.2171282691062582e-06 5.5920702707254974e-05 0.00031043754736771468 0.0010887195953766145 0.002995933980950227 0.0071555887773963782 0.015771491525200323 0.033804105852091329 0.076500887893677594 1 1
pdf.values = 0 0 0.00013122124829656863 0.001167162759255986 0.0044421984013840775 0.012096682222928267 0.027839371884979505 0.058806073487464187 0.12097828236031892 0.26011499456731058 0.69111441539656093 Inf 0
-sf.values = 1 1 0.999996782871731 0.999944079297293 0.999689562452632 0.998911280404623 0.997004066019050 0.992844411222604 0.984228508474800 0.966195894147909 0.923499112106322 0 0
+sf.values = 1 1 0.99999678287173088 0.99994407929729279 0.9996895624526323 0.9989112804046234 0.99700406601904978 0.99284441122260358 0.9842285084747997 0.96619589414790863 0.92349911210632241 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.22.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.22.properties
index 19381c2..749e340 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.22.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.22.properties
@@ -19,7 +19,7 @@ mean = 0.88888888888888884
variance = 0.017957351290684626
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 2.8511148630298162e-05 0.00047761405759400596 0.002544374616406807 0.0085162633709012919 0.022203904140477276 0.049735563119401985 0.10122151579386635 0.1950155281000755 0.37337491740225981 1 1
pdf.values = 0 0 0.0011529137302697221 0.0097827974015615871 0.035296594869406304 0.090369611411506442 0.19334951048069668 0.37354404860738988 0.68493846904682498 1.2521980673998829 2.5214223280998826 Inf 0
-sf.values = 1 1 0.999971488851370 0.999522385942406 0.997455625383593 0.991483736629099 0.977796095859523 0.950264436880598 0.898778484206134 0.804984471899925 0.626625082597740 0 0
+sf.values = 1 1 0.99997148885136966 0.99952238594240594 0.99745562538359323 0.99148373662909872 0.97779609585952276 0.95026443688059803 0.89877848420613371 0.8049844718999245 0.62662508259774019 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.23.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.23.properties
index cdc9362..b1e4667 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.23.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.23.properties
@@ -16,10 +16,10 @@
parameters = 4.0 1.0
# Computed using Matlab
mean = 0.80000000000000004
-variance = 0.026666666666666672
+variance = 0.026666666666666668
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.00010000000000000006 0.0016000000000000007 0.0080999999999999944 0.025600000000000001 0.062500000000000028 0.12959999999999997 0.24009999999999992 0.40960000000000019 0.65610000000000024 1 1
pdf.values = 0 0 0.0040000000000000018 0.032000000000000021 0.10799999999999996 0.25600000000000012 0.50000000000000022 0.8640000000000001 1.3719999999999999 2.0480000000000005 2.9160000000000008 4 0
-sf.values = 1 1 0.999900000000000 0.998400000000000 0.991900000000000 0.974400000000000 0.937500000000000 0.870400000000000 0.759900000000000 0.590400000000000 0.343900000000000 0 0
+sf.values = 1 1 0.99990000000000001 0.99839999999999995 0.9919 0.97440000000000004 0.9375 0.87040000000000006 0.75990000000000002 0.59039999999999981 0.34389999999999976 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.24.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.24.properties
index d782b4e..9c28704 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.24.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.24.properties
@@ -19,7 +19,7 @@ mean = 0.66666666666666663
variance = 0.031746031746031744
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.00046000000000000056 0.0067200000000000037 0.030779999999999971 0.087040000000000006 0.18749999999999992 0.33695999999999976 0.52822000000000002 0.73728000000000005 0.91854000000000002 1 1
pdf.values = 0 0 0.018000000000000016 0.12800000000000006 0.37799999999999967 0.7679999999999999 1.25 1.7279999999999998 2.0579999999999994 2.0479999999999996 1.4579999999999993 0 0
-sf.values = 1 1 0.999540000000000 0.993280000000000 0.969220000000000 0.912960000000000 0.812500000000000 0.663040000000000 0.471780000000000 0.262720000000000 0.0814600000000000 0 0
+sf.values = 1 1 0.99953999999999998 0.99327999999999994 0.96922000000000008 0.91295999999999999 0.81250000000000011 0.6630400000000003 0.47177999999999998 0.26271999999999995 0.081459999999999949 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.25.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.25.properties
index 8211a13..2515bf1 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.25.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.25.properties
@@ -19,7 +19,7 @@ mean = 0.5
variance = 0.027777777777777776
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
cdf.values = 0 0 0.0027280000000000021 0.033344000000000061 0.12603599999999995 0.28979200000000022 0.49999999999999967 0.71020799999999973 0.87396399999999996 0.96665599999999996 0.99727200000000005 1 1
pdf.values = 0 0 0.10206000000000011 0.57344000000000062 1.2965399999999996 1.9353600000000017 2.1875000000000018 1.9353600000000017 1.2965400000000007 0.57344000000000017 0.10206000000000001 0 0
-sf.values = 1 1 0.997272000000000 0.966656000000000 0.873964000000000 0.710208000000000 0.500000000000000 0.289792000000000 0.126036000000000 0.0333440000000000 0.00272800000000000 0 0
+sf.values = 1 1 0.99727200000000005 0.96665599999999996 0.87396400000000007 0.71020799999999973 0.50000000000000033 0.28979200000000022 0.12603600000000006 0.033343999999999999 0.0027279999999999974 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.3.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.3.properties
index f0a3f7d..b438150 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.3.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.3.properties
@@ -15,11 +15,11 @@
parameters = 0.1 1.0
# Computed using Matlab
-mean = 0.090909090909091
-variance = 0.039354584809130
+mean = 0.090909090909090912
+variance = 0.039354584809130261
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.794328234724282 0.851339922520785 0.886568150565213 0.912443536555481 0.933032991536807 0.950200216505676 0.964961095119818 0.977932768542929 0.989519258206214 1 1
-pdf.values = 0 Infinity 0.794328234724282 0.425669961260392 0.295522716855071 0.228110884138870 0.186606598307362 0.158366702750946 0.137851585017117 0.122241596067866 0.109946584245135 0.100000000000000 0
-sf.values = 1 1 0.205671765275719 0.148660077479215 0.113431849434787 0.0875564634445192 0.0669670084631926 0.0497997834943236 0.0350389048801824 0.0220672314570715 0.0104807417937856 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.79432823472428149 0.85133992252078461 0.88656815056521332 0.91244353655548083 0.93303299153680741 0.95020021650567643 0.96496109511981765 0.97793276854292854 0.98951925820621445 1 1
+pdf.values = 0 Inf 0.79432823472428149 0.42566996126039236 0.2955227168550712 0.22811088413887023 0.18660659830736154 0.15836670275094611 0.13785158501711683 0.1222415960678661 0.10994658424513494 0.10000000000000001 0
+sf.values = 1 1 0.20567176527571851 0.14866007747921539 0.11343184943478668 0.087556463444519186 0.066967008463192548 0.049799783494323582 0.035038904880182402 0.022067231457071495 0.010480741793785605 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.4.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.4.properties
index f0ba541..06c4822 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.4.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.4.properties
@@ -15,11 +15,11 @@
parameters = 0.1 2.0
# Computed using Matlab
-mean = 0.047619047619048
-variance = 0.014629507717065
+mean = 0.047619047619047616
+variance = 0.01462950771706532
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.865817775849467 0.919447116322448 0.948627921104778 0.967190148748810 0.979684641113648 0.988208225165904 0.993909927973412 0.997491423913787 0.999414450788277 1 1
-pdf.values = 0 Infinity 0.786384952377039 0.374589565909146 0.227552491978405 0.150553183531654 0.102633629069049 0.0696813492104163 0.0454910230556486 0.0268931511349305 0.0120941242669649 0 0
-sf.values = 1 1 0.134182224150533 0.0805528836775524 0.0513720788952217 0.0328098512511903 0.0203153588863522 0.0117917748340966 0.00609007202658787 0.00250857608621292 0.000585549211723463 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.86581777584946706 0.91944711632244758 0.94862792110477834 0.96719014874880971 0.97968464111364773 0.98820822516590345 0.99390992797341216 0.99749142391378709 0.99941445078827651 1 1
+pdf.values = 0 Inf 0.78638495237703898 0.37458956590914549 0.22755249197840488 0.15055318353165442 0.10263362906904888 0.069681349210416327 0.045491023055648561 0.026893151134930546 0.012094124266964851 0 0
+sf.values = 1 1 0.13418222415053294 0.080552883677552423 0.051372078895221711 0.032809851251190306 0.020315358886352223 0.011791774834096561 0.0060900720265878745 0.0025085760862129194 0.0005855492117234634 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.5.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.5.properties
index 9194496..07c4928 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.5.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.5.properties
@@ -15,11 +15,11 @@
parameters = 0.1 4.0
# Computed using Matlab
-mean = 0.024390243902439
-variance = 0.004665756844082
+mean = 0.024390243902439029
+variance = 0.0046657568440820708
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.923499112106323 0.966195894147909 0.984228508474800 0.992844411222604 0.997004066019050 0.998911280404623 0.999689562452632 0.999944079297293 0.999996782871731 1 1
-pdf.values = 0 Infinity 0.691114415396561 0.260114994567311 0.120978282360319 0.0588060734874642 0.0278393718849795 0.0120966822229283 0.00444219840138409 0.00116716275925599 0.000131221248296568 0 0
-sf.values = 1 1 0.0765008878936770 0.0338041058520913 0.0157714915252003 0.00715558877739638 0.00299593398095023 0.00108871959537661 0.000310437547367715 5.59207027072550e-05 3.21712826910625e-06 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.92349911210632296 0.96619589414790863 0.9842285084747997 0.99284441122260358 0.99700406601904978 0.9989112804046234 0.9996895624526323 0.99994407929729279 0.99999678287173088 1 1
+pdf.values = 0 Inf 0.6911144153965606 0.26011499456731052 0.12097828236031892 0.058806073487464187 0.027839371884979505 0.012096682222928267 0.0044421984013840853 0.0011671627592559851 0.00013122124829656841 0 0
+sf.values = 1 1 0.076500887893677039 0.033804105852091329 0.015771491525200323 0.0071555887773963782 0.002995933980950227 0.0010887195953766145 0.00031043754736771479 5.5920702707254981e-05 3.2171282691062514e-06 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.6.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.6.properties
index aeaacde..9ec6aff 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.6.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.6.properties
@@ -15,11 +15,11 @@
parameters = 0.5 0.1
# Computed using Matlab
-mean = 0.833333333333333
-variance = 0.086805555555556
+mean = 0.83333333333333337
+variance = 0.086805555555555566
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.0576337116842096 0.0843593596249964 0.107341412163594 0.129396563019995 0.151998287600230 0.176505161463343 0.204823469632632 0.240695780554681 0.295166377948779 1 1
-pdf.values = 0 Infinity 0.307055512274369 0.241401058813832 0.222272786172717 0.221140881157234 0.233065049073820 0.260079151351645 0.311944266317603 0.420303655461325 0.739458664410645 Infinity 0
-sf.values = 1 1 0.942366288315790 0.915640640375004 0.892658587836406 0.870603436980005 0.848001712399770 0.823494838536657 0.795176530367368 0.759304219445319 0.704833622051221 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.057633711684209617 0.084359359624996397 0.10734141216359447 0.12939656301999489 0.15199828760022974 0.17650516146334294 0.20482346963263176 0.24069578055468144 0.29516637794877887 1 1
+pdf.values = 0 Inf 0.30705551227436922 0.24140105881383231 0.22227278617271662 0.22114088115723446 0.23306504907381986 0.26007915135164533 0.31194426631760319 0.42030365546132514 0.73945866441064467 Inf 0
+sf.values = 1 1 0.94236628831579039 0.91564064037500359 0.8926585878364055 0.87060343698000509 0.84800171239977029 0.82349483853665706 0.79517653036736824 0.75930421944531856 0.70483362205122113 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.7.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.7.properties
index d4c3cc0..281a1b0 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.7.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.7.properties
@@ -19,7 +19,7 @@ mean = 0.5
variance = 0.125
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.204832764699133 0.295167235300867 0.369010119565545 0.435905783151025 0.500000000000000 0.564094216848975 0.630989880434455 0.704832764699133 0.795167235300867 1 1
-pdf.values = 0 Infinity 1.06103295394597 0.795774715459477 0.694609118042857 0.649747334361397 0.636619772367582 0.649747334361397 0.694609118042857 0.795774715459477 1.06103295394597 Infinity 0
-sf.values = 1 1 0.795167235300867 0.704832764699133 0.630989880434455 0.564094216848975 0.500000000000000 0.435905783151025 0.369010119565545 0.295167235300867 0.204832764699133 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.20483276469913342 0.29516723530086691 0.3690101195655453 0.43590578315102529 0.50000000000000011 0.56409421684897465 0.6309898804344547 0.70483276469913303 0.79516723530086675 1 1
+pdf.values = 0 Inf 1.0610329539459691 0.79577471545947676 0.69460911804285685 0.64974733436139687 0.63661977236758149 0.64974733436139687 0.69460911804285674 0.79577471545947687 1.0610329539459693 Inf 0
+sf.values = 1 1 0.79516723530086653 0.70483276469913303 0.6309898804344547 0.56409421684897465 0.49999999999999989 0.43590578315102529 0.36901011956554536 0.29516723530086691 0.20483276469913331 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.8.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.8.properties
index 5f1c437..ff9d4a3 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.8.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.8.properties
@@ -15,11 +15,11 @@
parameters = 0.5 1.0
# Computed using Matlab
-mean = 0.333333333333333
-variance = 0.088888888888889
+mean = 0.33333333333333331
+variance = 0.088888888888888892
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.316227766016838 0.447213595499958 0.547722557505166 0.632455532033676 0.707106781186547 0.774596669241483 0.836660026534075 0.894427190999916 0.948683298050514 1 1
-pdf.values = 0 Infinity 1.58113883008419 1.11803398874990 0.912870929175277 0.790569415042095 0.707106781186548 0.645497224367903 0.597614304667197 0.559016994374948 0.527046276694730 0.500000000000000 0
-sf.values = 1 1 0.683772233983162 0.552786404500042 0.452277442494834 0.367544467966324 0.292893218813453 0.225403330758517 0.163339973465925 0.105572809000084 0.0513167019494862 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.31622776601683805 0.44721359549995793 0.54772255750516619 0.63245553203367599 0.70710678118654724 0.7745966692414834 0.83666002653407534 0.89442719099991586 0.94868329805051377 1 1
+pdf.values = 0 Inf 1.5811388300841898 1.1180339887498949 0.91287092917527701 0.79056941504209488 0.70710678118654768 0.64549722436790291 0.59761430466719689 0.55901699437494745 0.52704627669472992 0.5 0
+sf.values = 1 1 0.683772233983162 0.55278640450004213 0.45227744249483381 0.36754446796632401 0.2928932188134527 0.22540333075851654 0.16333997346592463 0.1055728090000841 0.051316701949486225 0 0
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.9.properties b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.9.properties
index 4f096f6..2cd8b57 100644
--- a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.9.properties
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/test.beta.9.properties
@@ -15,11 +15,11 @@
parameters = 0.5 2.0
# Computed using Matlab
-mean = 0.200000000000000
-variance = 0.045714285714286
+mean = 0.20000000000000001
+variance = 0.045714285714285714
lower = 0
upper = 1
-cdf.points = -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1
-cdf.values = 0 0 0.458530260724415 0.626099033699941 0.739425452631974 0.822192191643779 0.883883476483184 0.929516003089780 0.962159030514187 0.983869910099907 0.996117462953040 1 1
-pdf.values = 0 Inf 2.13453742061366 1.34164078649987 0.958514475634041 0.711512473537885 0.530330085889911 0.387298334620742 0.268926437100239 0.167705098312484 0.0790569415042095 0 0
-sf.values = 1 1 0.541469739275585 0.373900966300059 0.260574547368026 0.177807808356221 0.116116523516816 0.0704839969102200 0.0378409694858131 0.0161300899000925 0.00388253704696051 0 0
+cdf.points = -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
+cdf.values = 0 0 0.45853026072441505 0.62609903369994102 0.73942545263197434 0.82219219164377877 0.88388347648318444 0.92951600308978 0.96215903051418694 0.98386991009990743 0.99611746295303949 1 1
+pdf.values = 0 Inf 2.1345374206136558 1.3416407864998741 0.9585144756340408 0.71151247353788538 0.53033008588991071 0.38729833462074181 0.26892643710023867 0.16770509831248423 0.079056941504209458 0 0
+sf.values = 1 1 0.54146973927558495 0.37390096630005898 0.26057454736802566 0.17780780835622126 0.1161165235168156 0.070483996910219962 0.037840969485813107 0.016130089900092535 0.0038825370469605072 0 0