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+<title>AbstractDiscreteDistribution xref</title>
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+<div id="overview"><a href="../../../../../../apidocs/org/apache/commons/statistics/distribution/AbstractDiscreteDistribution.html">View Javadoc</a></div><pre>
+<a class="jxr_linenumber" name="L1" href="#L1">1</a> <em class="jxr_comment">/*</em>
+<a class="jxr_linenumber" name="L2" href="#L2">2</a> <em class="jxr_comment"> * Licensed to the Apache Software Foundation (ASF) under one or more</em>
+<a class="jxr_linenumber" name="L3" href="#L3">3</a> <em class="jxr_comment"> * contributor license agreements. See the NOTICE file distributed with</em>
+<a class="jxr_linenumber" name="L4" href="#L4">4</a> <em class="jxr_comment"> * this work for additional information regarding copyright ownership.</em>
+<a class="jxr_linenumber" name="L5" href="#L5">5</a> <em class="jxr_comment"> * The ASF licenses this file to You under the Apache License, Version 2.0</em>
+<a class="jxr_linenumber" name="L6" href="#L6">6</a> <em class="jxr_comment"> * (the "License"); you may not use this file except in compliance with</em>
+<a class="jxr_linenumber" name="L7" href="#L7">7</a> <em class="jxr_comment"> * the License. You may obtain a copy of the License at</em>
+<a class="jxr_linenumber" name="L8" href="#L8">8</a> <em class="jxr_comment"> *</em>
+<a class="jxr_linenumber" name="L9" href="#L9">9</a> <em class="jxr_comment"> * <a href="http://www.apache.org/licenses/LICENSE-2.0" target="alexandria_uri">http://www.apache.org/licenses/LICENSE-2.0</a></em>
+<a class="jxr_linenumber" name="L10" href="#L10">10</a> <em class="jxr_comment"> *</em>
+<a class="jxr_linenumber" name="L11" href="#L11">11</a> <em class="jxr_comment"> * Unless required by applicable law or agreed to in writing, software</em>
+<a class="jxr_linenumber" name="L12" href="#L12">12</a> <em class="jxr_comment"> * distributed under the License is distributed on an "AS IS" BASIS,</em>
+<a class="jxr_linenumber" name="L13" href="#L13">13</a> <em class="jxr_comment"> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</em>
+<a class="jxr_linenumber" name="L14" href="#L14">14</a> <em class="jxr_comment"> * See the License for the specific language governing permissions and</em>
+<a class="jxr_linenumber" name="L15" href="#L15">15</a> <em class="jxr_comment"> * limitations under the License.</em>
+<a class="jxr_linenumber" name="L16" href="#L16">16</a> <em class="jxr_comment"> */</em>
+<a class="jxr_linenumber" name="L17" href="#L17">17</a> <strong class="jxr_keyword">package</strong> org.apache.commons.statistics.distribution;
+<a class="jxr_linenumber" name="L18" href="#L18">18</a>
+<a class="jxr_linenumber" name="L19" href="#L19">19</a> <strong class="jxr_keyword">import</strong> java.util.function.IntUnaryOperator;
+<a class="jxr_linenumber" name="L20" href="#L20">20</a> <strong class="jxr_keyword">import</strong> org.apache.commons.rng.UniformRandomProvider;
+<a class="jxr_linenumber" name="L21" href="#L21">21</a> <strong class="jxr_keyword">import</strong> org.apache.commons.rng.sampling.distribution.InverseTransformDiscreteSampler;
+<a class="jxr_linenumber" name="L22" href="#L22">22</a>
+<a class="jxr_linenumber" name="L23" href="#L23">23</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L24" href="#L24">24</a> <em class="jxr_javadoccomment"> * Base class for integer-valued discrete distributions. Default</em>
+<a class="jxr_linenumber" name="L25" href="#L25">25</a> <em class="jxr_javadoccomment"> * implementations are provided for some of the methods that do not vary</em>
+<a class="jxr_linenumber" name="L26" href="#L26">26</a> <em class="jxr_javadoccomment"> * from distribution to distribution.</em>
+<a class="jxr_linenumber" name="L27" href="#L27">27</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L28" href="#L28">28</a> <em class="jxr_javadoccomment"> * <p>This base class provides a default factory method for creating</em>
+<a class="jxr_linenumber" name="L29" href="#L29">29</a> <em class="jxr_javadoccomment"> * a {@link DiscreteDistribution.Sampler sampler instance} that uses the</em>
+<a class="jxr_linenumber" name="L30" href="#L30">30</a> <em class="jxr_javadoccomment"> * <a href="<a href="https://en.wikipedia.org/wiki/Inverse_transform_sampling" target="alexandria_uri">https://en.wikipedia.org/wiki/Inverse_transform_sampling</a>"></em>
+<a class="jxr_linenumber" name="L31" href="#L31">31</a> <em class="jxr_javadoccomment"> * inversion method</a> for generating random samples that follow the</em>
+<a class="jxr_linenumber" name="L32" href="#L32">32</a> <em class="jxr_javadoccomment"> * distribution.</em>
+<a class="jxr_linenumber" name="L33" href="#L33">33</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L34" href="#L34">34</a> <em class="jxr_javadoccomment"> * <p>The class provides functionality to evaluate the probability in a range</em>
+<a class="jxr_linenumber" name="L35" href="#L35">35</a> <em class="jxr_javadoccomment"> * using either the cumulative probability or the survival probability.</em>
+<a class="jxr_linenumber" name="L36" href="#L36">36</a> <em class="jxr_javadoccomment"> * The survival probability is used if both arguments to</em>
+<a class="jxr_linenumber" name="L37" href="#L37">37</a> <em class="jxr_javadoccomment"> * {@link #probability(int, int)} are above the median.</em>
+<a class="jxr_linenumber" name="L38" href="#L38">38</a> <em class="jxr_javadoccomment"> * Child classes with a known median can override the default {@link #getMedian()}</em>
+<a class="jxr_linenumber" name="L39" href="#L39">39</a> <em class="jxr_javadoccomment"> * method.</em>
+<a class="jxr_linenumber" name="L40" href="#L40">40</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L41" href="#L41">41</a> <strong class="jxr_keyword">abstract</strong> <strong class="jxr_keyword">class</strong> <a name="AbstractDiscreteDistribution" href="../../../../../org/apache/commons/statistics/distribution/AbstractDiscreteDistribution.html#AbstractDiscreteDistribution">AbstractDiscreteDistribution</a>
+<a class="jxr_linenumber" name="L42" href="#L42">42</a> <strong class="jxr_keyword">implements</strong> <a name="DiscreteDistribution" href="../../../../../org/apache/commons/statistics/distribution/DiscreteDistribution.html#DiscreteDistribution">DiscreteDistribution</a> {
+<a class="jxr_linenumber" name="L43" href="#L43">43</a> <em class="jxr_javadoccomment">/** Marker value for no median.</em>
+<a class="jxr_linenumber" name="L44" href="#L44">44</a> <em class="jxr_javadoccomment"> * This is a long to be outside the value of any possible int valued median. */</em>
+<a class="jxr_linenumber" name="L45" href="#L45">45</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">long</strong> NO_MEDIAN = Long.MIN_VALUE;
+<a class="jxr_linenumber" name="L46" href="#L46">46</a>
+<a class="jxr_linenumber" name="L47" href="#L47">47</a> <em class="jxr_javadoccomment">/** Cached value of the median. */</em>
+<a class="jxr_linenumber" name="L48" href="#L48">48</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">long</strong> median = NO_MEDIAN;
+<a class="jxr_linenumber" name="L49" href="#L49">49</a>
+<a class="jxr_linenumber" name="L50" href="#L50">50</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L51" href="#L51">51</a> <em class="jxr_javadoccomment"> * Gets the median. This is used to determine if the arguments to the</em>
+<a class="jxr_linenumber" name="L52" href="#L52">52</a> <em class="jxr_javadoccomment"> * {@link #probability(int, int)} function are in the upper or lower domain.</em>
+<a class="jxr_linenumber" name="L53" href="#L53">53</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L54" href="#L54">54</a> <em class="jxr_javadoccomment"> * <p>The default implementation calls {@link #inverseCumulativeProbability(double)}</em>
+<a class="jxr_linenumber" name="L55" href="#L55">55</a> <em class="jxr_javadoccomment"> * with a value of 0.5.</em>
+<a class="jxr_linenumber" name="L56" href="#L56">56</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L57" href="#L57">57</a> <em class="jxr_javadoccomment"> * @return the median</em>
+<a class="jxr_linenumber" name="L58" href="#L58">58</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L59" href="#L59">59</a> <strong class="jxr_keyword">int</strong> getMedian() {
+<a class="jxr_linenumber" name="L60" href="#L60">60</a> <strong class="jxr_keyword">long</strong> m = median;
+<a class="jxr_linenumber" name="L61" href="#L61">61</a> <strong class="jxr_keyword">if</strong> (m == NO_MEDIAN) {
+<a class="jxr_linenumber" name="L62" href="#L62">62</a> median = m = inverseCumulativeProbability(0.5);
+<a class="jxr_linenumber" name="L63" href="#L63">63</a> }
+<a class="jxr_linenumber" name="L64" href="#L64">64</a> <strong class="jxr_keyword">return</strong> (<strong class="jxr_keyword">int</strong>) m;
+<a class="jxr_linenumber" name="L65" href="#L65">65</a> }
+<a class="jxr_linenumber" name="L66" href="#L66">66</a>
+<a class="jxr_linenumber" name="L67" href="#L67">67</a> <em class="jxr_javadoccomment">/** {@inheritDoc} */</em>
+<a class="jxr_linenumber" name="L68" href="#L68">68</a> @Override
+<a class="jxr_linenumber" name="L69" href="#L69">69</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> probability(<strong class="jxr_keyword">int</strong> x0,
+<a class="jxr_linenumber" name="L70" href="#L70">70</a> <strong class="jxr_keyword">int</strong> x1) {
+<a class="jxr_linenumber" name="L71" href="#L71">71</a> <strong class="jxr_keyword">if</strong> (x0 > x1) {
+<a class="jxr_linenumber" name="L72" href="#L72">72</a> <strong class="jxr_keyword">throw</strong> <strong class="jxr_keyword">new</strong> <a name="DistributionException" href="../../../../../org/apache/commons/statistics/distribution/DistributionException.html#DistributionException">DistributionException</a>(DistributionException.INVALID_RANGE_LOW_GT_HIGH, x0, x1);
+<a class="jxr_linenumber" name="L73" href="#L73">73</a> }
+<a class="jxr_linenumber" name="L74" href="#L74">74</a> <em class="jxr_comment">// As per the default interface method handle special cases:</em>
+<a class="jxr_linenumber" name="L75" href="#L75">75</a> <em class="jxr_comment">// x0 = x1 : return 0</em>
+<a class="jxr_linenumber" name="L76" href="#L76">76</a> <em class="jxr_comment">// x0 + 1 = x1 : return probability(x1)</em>
+<a class="jxr_linenumber" name="L77" href="#L77">77</a> <em class="jxr_comment">// Long addition avoids overflow</em>
+<a class="jxr_linenumber" name="L78" href="#L78">78</a> <strong class="jxr_keyword">if</strong> (x0 + 1L >= x1) {
+<a class="jxr_linenumber" name="L79" href="#L79">79</a> <strong class="jxr_keyword">return</strong> x0 == x1 ? 0.0 : probability(x1);
+<a class="jxr_linenumber" name="L80" href="#L80">80</a> }
+<a class="jxr_linenumber" name="L81" href="#L81">81</a>
+<a class="jxr_linenumber" name="L82" href="#L82">82</a> <em class="jxr_comment">// Use the survival probability when in the upper domain [3]:</em>
+<a class="jxr_linenumber" name="L83" href="#L83">83</a> <em class="jxr_comment">//</em>
+<a class="jxr_linenumber" name="L84" href="#L84">84</a> <em class="jxr_comment">// lower median upper</em>
+<a class="jxr_linenumber" name="L85" href="#L85">85</a> <em class="jxr_comment">// | | |</em>
+<a class="jxr_linenumber" name="L86" href="#L86">86</a> <em class="jxr_comment">// 1. |------|</em>
+<a class="jxr_linenumber" name="L87" href="#L87">87</a> <em class="jxr_comment">// x0 x1</em>
+<a class="jxr_linenumber" name="L88" href="#L88">88</a> <em class="jxr_comment">// 2. |----------|</em>
+<a class="jxr_linenumber" name="L89" href="#L89">89</a> <em class="jxr_comment">// x0 x1</em>
+<a class="jxr_linenumber" name="L90" href="#L90">90</a> <em class="jxr_comment">// 3. |--------|</em>
+<a class="jxr_linenumber" name="L91" href="#L91">91</a> <em class="jxr_comment">// x0 x1</em>
+<a class="jxr_linenumber" name="L92" href="#L92">92</a>
+<a class="jxr_linenumber" name="L93" href="#L93">93</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> m = getMedian();
+<a class="jxr_linenumber" name="L94" href="#L94">94</a> <strong class="jxr_keyword">if</strong> (x0 >= m) {
+<a class="jxr_linenumber" name="L95" href="#L95">95</a> <strong class="jxr_keyword">return</strong> survivalProbability(x0) - survivalProbability(x1);
+<a class="jxr_linenumber" name="L96" href="#L96">96</a> }
+<a class="jxr_linenumber" name="L97" href="#L97">97</a> <strong class="jxr_keyword">return</strong> cumulativeProbability(x1) - cumulativeProbability(x0);
+<a class="jxr_linenumber" name="L98" href="#L98">98</a> }
+<a class="jxr_linenumber" name="L99" href="#L99">99</a>
+<a class="jxr_linenumber" name="L100" href="#L100">100</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L101" href="#L101">101</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L102" href="#L102">102</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L103" href="#L103">103</a> <em class="jxr_javadoccomment"> * <p>The default implementation returns:</em>
+<a class="jxr_linenumber" name="L104" href="#L104">104</a> <em class="jxr_javadoccomment"> * <ul></em>
+<a class="jxr_linenumber" name="L105" href="#L105">105</a> <em class="jxr_javadoccomment"> * <li>{@link #getSupportLowerBound()} for {@code p = 0},</li></em>
+<a class="jxr_linenumber" name="L106" href="#L106">106</a> <em class="jxr_javadoccomment"> * <li>{@link #getSupportUpperBound()} for {@code p = 1}, or</li></em>
+<a class="jxr_linenumber" name="L107" href="#L107">107</a> <em class="jxr_javadoccomment"> * <li>the result of a binary search between the lower and upper bound using</em>
+<a class="jxr_linenumber" name="L108" href="#L108">108</a> <em class="jxr_javadoccomment"> * {@link #cumulativeProbability(int) cumulativeProbability(x)}.</em>
+<a class="jxr_linenumber" name="L109" href="#L109">109</a> <em class="jxr_javadoccomment"> * The bounds may be bracketed for efficiency.</li></em>
+<a class="jxr_linenumber" name="L110" href="#L110">110</a> <em class="jxr_javadoccomment"> * </ul></em>
+<a class="jxr_linenumber" name="L111" href="#L111">111</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L112" href="#L112">112</a> <em class="jxr_javadoccomment"> * @throws IllegalArgumentException if {@code p < 0} or {@code p > 1}</em>
+<a class="jxr_linenumber" name="L113" href="#L113">113</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L114" href="#L114">114</a> @Override
+<a class="jxr_linenumber" name="L115" href="#L115">115</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">int</strong> inverseCumulativeProbability(<strong class="jxr_keyword">double</strong> p) {
+<a class="jxr_linenumber" name="L116" href="#L116">116</a> ArgumentUtils.checkProbability(p);
+<a class="jxr_linenumber" name="L117" href="#L117">117</a> <strong class="jxr_keyword">return</strong> inverseProbability(p, 1 - p, false);
+<a class="jxr_linenumber" name="L118" href="#L118">118</a> }
+<a class="jxr_linenumber" name="L119" href="#L119">119</a>
+<a class="jxr_linenumber" name="L120" href="#L120">120</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L121" href="#L121">121</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L122" href="#L122">122</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L123" href="#L123">123</a> <em class="jxr_javadoccomment"> * <p>The default implementation returns:</em>
+<a class="jxr_linenumber" name="L124" href="#L124">124</a> <em class="jxr_javadoccomment"> * <ul></em>
+<a class="jxr_linenumber" name="L125" href="#L125">125</a> <em class="jxr_javadoccomment"> * <li>{@link #getSupportLowerBound()} for {@code p = 1},</li></em>
+<a class="jxr_linenumber" name="L126" href="#L126">126</a> <em class="jxr_javadoccomment"> * <li>{@link #getSupportUpperBound()} for {@code p = 0}, or</li></em>
+<a class="jxr_linenumber" name="L127" href="#L127">127</a> <em class="jxr_javadoccomment"> * <li>the result of a binary search between the lower and upper bound using</em>
+<a class="jxr_linenumber" name="L128" href="#L128">128</a> <em class="jxr_javadoccomment"> * {@link #survivalProbability(int) survivalProbability(x)}.</em>
+<a class="jxr_linenumber" name="L129" href="#L129">129</a> <em class="jxr_javadoccomment"> * The bounds may be bracketed for efficiency.</li></em>
+<a class="jxr_linenumber" name="L130" href="#L130">130</a> <em class="jxr_javadoccomment"> * </ul></em>
+<a class="jxr_linenumber" name="L131" href="#L131">131</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L132" href="#L132">132</a> <em class="jxr_javadoccomment"> * @throws IllegalArgumentException if {@code p < 0} or {@code p > 1}</em>
+<a class="jxr_linenumber" name="L133" href="#L133">133</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L134" href="#L134">134</a> @Override
+<a class="jxr_linenumber" name="L135" href="#L135">135</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">int</strong> inverseSurvivalProbability(<strong class="jxr_keyword">double</strong> p) {
+<a class="jxr_linenumber" name="L136" href="#L136">136</a> ArgumentUtils.checkProbability(p);
+<a class="jxr_linenumber" name="L137" href="#L137">137</a> <strong class="jxr_keyword">return</strong> inverseProbability(1 - p, p, <strong class="jxr_keyword">true</strong>);
+<a class="jxr_linenumber" name="L138" href="#L138">138</a> }
+<a class="jxr_linenumber" name="L139" href="#L139">139</a>
+<a class="jxr_linenumber" name="L140" href="#L140">140</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L141" href="#L141">141</a> <em class="jxr_javadoccomment"> * Implementation for the inverse cumulative or survival probability.</em>
+<a class="jxr_linenumber" name="L142" href="#L142">142</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L143" href="#L143">143</a> <em class="jxr_javadoccomment"> * @param p Cumulative probability.</em>
+<a class="jxr_linenumber" name="L144" href="#L144">144</a> <em class="jxr_javadoccomment"> * @param q Survival probability.</em>
+<a class="jxr_linenumber" name="L145" href="#L145">145</a> <em class="jxr_javadoccomment"> * @param complement Set to true to compute the inverse survival probability</em>
+<a class="jxr_linenumber" name="L146" href="#L146">146</a> <em class="jxr_javadoccomment"> * @return the value</em>
+<a class="jxr_linenumber" name="L147" href="#L147">147</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L148" href="#L148">148</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">int</strong> inverseProbability(<strong class="jxr_keyword">double</strong> p, <strong class="jxr_keyword">double</strong> q, <strong class="jxr_keyword">boolean</strong> complement) {
+<a class="jxr_linenumber" name="L149" href="#L149">149</a>
+<a class="jxr_linenumber" name="L150" href="#L150">150</a> <strong class="jxr_keyword">int</strong> lower = getSupportLowerBound();
+<a class="jxr_linenumber" name="L151" href="#L151">151</a> <strong class="jxr_keyword">if</strong> (p == 0) {
+<a class="jxr_linenumber" name="L152" href="#L152">152</a> <strong class="jxr_keyword">return</strong> lower;
+<a class="jxr_linenumber" name="L153" href="#L153">153</a> }
+<a class="jxr_linenumber" name="L154" href="#L154">154</a> <strong class="jxr_keyword">int</strong> upper = getSupportUpperBound();
+<a class="jxr_linenumber" name="L155" href="#L155">155</a> <strong class="jxr_keyword">if</strong> (q == 0) {
+<a class="jxr_linenumber" name="L156" href="#L156">156</a> <strong class="jxr_keyword">return</strong> upper;
+<a class="jxr_linenumber" name="L157" href="#L157">157</a> }
+<a class="jxr_linenumber" name="L158" href="#L158">158</a>
+<a class="jxr_linenumber" name="L159" href="#L159">159</a> <em class="jxr_comment">// The binary search sets the upper value to the mid-point</em>
+<a class="jxr_linenumber" name="L160" href="#L160">160</a> <em class="jxr_comment">// based on fun(x) >= 0. The upper value is returned.</em>
+<a class="jxr_linenumber" name="L161" href="#L161">161</a> <em class="jxr_comment">//</em>
+<a class="jxr_linenumber" name="L162" href="#L162">162</a> <em class="jxr_comment">// Create a function to search for x where the upper bound can be</em>
+<a class="jxr_linenumber" name="L163" href="#L163">163</a> <em class="jxr_comment">// lowered if:</em>
+<a class="jxr_linenumber" name="L164" href="#L164">164</a> <em class="jxr_comment">// cdf(x) >= p</em>
+<a class="jxr_linenumber" name="L165" href="#L165">165</a> <em class="jxr_comment">// sf(x) <= q</em>
+<a class="jxr_linenumber" name="L166" href="#L166">166</a> <strong class="jxr_keyword">final</strong> IntUnaryOperator fun = complement ?
+<a class="jxr_linenumber" name="L167" href="#L167">167</a> x -> Double.compare(q, survivalProbability(x)) :
+<a class="jxr_linenumber" name="L168" href="#L168">168</a> x -> Double.compare(cumulativeProbability(x), p);
+<a class="jxr_linenumber" name="L169" href="#L169">169</a>
+<a class="jxr_linenumber" name="L170" href="#L170">170</a> <strong class="jxr_keyword">if</strong> (lower == Integer.MIN_VALUE) {
+<a class="jxr_linenumber" name="L171" href="#L171">171</a> <strong class="jxr_keyword">if</strong> (fun.applyAsInt(lower) >= 0) {
+<a class="jxr_linenumber" name="L172" href="#L172">172</a> <strong class="jxr_keyword">return</strong> lower;
+<a class="jxr_linenumber" name="L173" href="#L173">173</a> }
+<a class="jxr_linenumber" name="L174" href="#L174">174</a> } <strong class="jxr_keyword">else</strong> {
+<a class="jxr_linenumber" name="L175" href="#L175">175</a> <em class="jxr_comment">// this ensures:</em>
+<a class="jxr_linenumber" name="L176" href="#L176">176</a> <em class="jxr_comment">// cumulativeProbability(lower) < p</em>
+<a class="jxr_linenumber" name="L177" href="#L177">177</a> <em class="jxr_comment">// survivalProbability(lower) > q</em>
+<a class="jxr_linenumber" name="L178" href="#L178">178</a> <em class="jxr_comment">// which is important for the solving step</em>
+<a class="jxr_linenumber" name="L179" href="#L179">179</a> lower -= 1;
+<a class="jxr_linenumber" name="L180" href="#L180">180</a> }
+<a class="jxr_linenumber" name="L181" href="#L181">181</a>
+<a class="jxr_linenumber" name="L182" href="#L182">182</a> <em class="jxr_comment">// use the one-sided Chebyshev inequality to narrow the bracket</em>
+<a class="jxr_linenumber" name="L183" href="#L183">183</a> <em class="jxr_comment">// cf. AbstractContinuousDistribution.inverseCumulativeProbability(double)</em>
+<a class="jxr_linenumber" name="L184" href="#L184">184</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> mu = getMean();
+<a class="jxr_linenumber" name="L185" href="#L185">185</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> sig = Math.sqrt(getVariance());
+<a class="jxr_linenumber" name="L186" href="#L186">186</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">boolean</strong> chebyshevApplies = Double.isFinite(mu) &&
+<a class="jxr_linenumber" name="L187" href="#L187">187</a> ArgumentUtils.isFiniteStrictlyPositive(sig);
+<a class="jxr_linenumber" name="L188" href="#L188">188</a>
+<a class="jxr_linenumber" name="L189" href="#L189">189</a> <strong class="jxr_keyword">if</strong> (chebyshevApplies) {
+<a class="jxr_linenumber" name="L190" href="#L190">190</a> <strong class="jxr_keyword">double</strong> tmp = mu - sig * Math.sqrt(q / p);
+<a class="jxr_linenumber" name="L191" href="#L191">191</a> <strong class="jxr_keyword">if</strong> (tmp > lower) {
+<a class="jxr_linenumber" name="L192" href="#L192">192</a> lower = ((<strong class="jxr_keyword">int</strong>) Math.ceil(tmp)) - 1;
+<a class="jxr_linenumber" name="L193" href="#L193">193</a> }
+<a class="jxr_linenumber" name="L194" href="#L194">194</a> tmp = mu + sig * Math.sqrt(p / q);
+<a class="jxr_linenumber" name="L195" href="#L195">195</a> <strong class="jxr_keyword">if</strong> (tmp < upper) {
+<a class="jxr_linenumber" name="L196" href="#L196">196</a> upper = ((<strong class="jxr_keyword">int</strong>) Math.ceil(tmp)) - 1;
+<a class="jxr_linenumber" name="L197" href="#L197">197</a> }
+<a class="jxr_linenumber" name="L198" href="#L198">198</a> }
+<a class="jxr_linenumber" name="L199" href="#L199">199</a>
+<a class="jxr_linenumber" name="L200" href="#L200">200</a> <strong class="jxr_keyword">return</strong> solveInverseProbability(fun, lower, upper);
+<a class="jxr_linenumber" name="L201" href="#L201">201</a> }
+<a class="jxr_linenumber" name="L202" href="#L202">202</a>
+<a class="jxr_linenumber" name="L203" href="#L203">203</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L204" href="#L204">204</a> <em class="jxr_javadoccomment"> * This is a utility function used by {@link</em>
+<a class="jxr_linenumber" name="L205" href="#L205">205</a> <em class="jxr_javadoccomment"> * #inverseProbability(double, double, boolean)}. It assumes</em>
+<a class="jxr_linenumber" name="L206" href="#L206">206</a> <em class="jxr_javadoccomment"> * that the inverse probability lies in the bracket {@code</em>
+<a class="jxr_linenumber" name="L207" href="#L207">207</a> <em class="jxr_javadoccomment"> * (lower, upper]}. The implementation does simple bisection to find the</em>
+<a class="jxr_linenumber" name="L208" href="#L208">208</a> <em class="jxr_javadoccomment"> * smallest {@code x} such that {@code fun(x) >= 0}.</em>
+<a class="jxr_linenumber" name="L209" href="#L209">209</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L210" href="#L210">210</a> <em class="jxr_javadoccomment"> * @param fun Probability function.</em>
+<a class="jxr_linenumber" name="L211" href="#L211">211</a> <em class="jxr_javadoccomment"> * @param lowerBound Value satisfying {@code fun(lower) < 0}.</em>
+<a class="jxr_linenumber" name="L212" href="#L212">212</a> <em class="jxr_javadoccomment"> * @param upperBound Value satisfying {@code fun(upper) >= 0}.</em>
+<a class="jxr_linenumber" name="L213" href="#L213">213</a> <em class="jxr_javadoccomment"> * @return the smallest x</em>
+<a class="jxr_linenumber" name="L214" href="#L214">214</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L215" href="#L215">215</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">int</strong> solveInverseProbability(IntUnaryOperator fun,
+<a class="jxr_linenumber" name="L216" href="#L216">216</a> <strong class="jxr_keyword">int</strong> lowerBound,
+<a class="jxr_linenumber" name="L217" href="#L217">217</a> <strong class="jxr_keyword">int</strong> upperBound) {
+<a class="jxr_linenumber" name="L218" href="#L218">218</a> <em class="jxr_comment">// Use long to prevent overflow during computation of the middle</em>
+<a class="jxr_linenumber" name="L219" href="#L219">219</a> <strong class="jxr_keyword">long</strong> lower = lowerBound;
+<a class="jxr_linenumber" name="L220" href="#L220">220</a> <strong class="jxr_keyword">long</strong> upper = upperBound;
+<a class="jxr_linenumber" name="L221" href="#L221">221</a> <strong class="jxr_keyword">while</strong> (lower + 1 < upper) {
+<a class="jxr_linenumber" name="L222" href="#L222">222</a> <em class="jxr_comment">// Note: Cannot replace division by 2 with a right shift because</em>
+<a class="jxr_linenumber" name="L223" href="#L223">223</a> <em class="jxr_comment">// (lower + upper) can be negative.</em>
+<a class="jxr_linenumber" name="L224" href="#L224">224</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">long</strong> middle = (lower + upper) / 2;
+<a class="jxr_linenumber" name="L225" href="#L225">225</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">int</strong> pm = fun.applyAsInt((<strong class="jxr_keyword">int</strong>) middle);
+<a class="jxr_linenumber" name="L226" href="#L226">226</a> <strong class="jxr_keyword">if</strong> (pm < 0) {
+<a class="jxr_linenumber" name="L227" href="#L227">227</a> lower = middle;
+<a class="jxr_linenumber" name="L228" href="#L228">228</a> } <strong class="jxr_keyword">else</strong> {
+<a class="jxr_linenumber" name="L229" href="#L229">229</a> upper = middle;
+<a class="jxr_linenumber" name="L230" href="#L230">230</a> }
+<a class="jxr_linenumber" name="L231" href="#L231">231</a> }
+<a class="jxr_linenumber" name="L232" href="#L232">232</a> <strong class="jxr_keyword">return</strong> (<strong class="jxr_keyword">int</strong>) upper;
+<a class="jxr_linenumber" name="L233" href="#L233">233</a> }
+<a class="jxr_linenumber" name="L234" href="#L234">234</a>
+<a class="jxr_linenumber" name="L235" href="#L235">235</a> <em class="jxr_javadoccomment">/** {@inheritDoc} */</em>
+<a class="jxr_linenumber" name="L236" href="#L236">236</a> @Override
+<a class="jxr_linenumber" name="L237" href="#L237">237</a> <strong class="jxr_keyword">public</strong> DiscreteDistribution.Sampler createSampler(<strong class="jxr_keyword">final</strong> UniformRandomProvider rng) {
+<a class="jxr_linenumber" name="L238" href="#L238">238</a> <em class="jxr_comment">// Inversion method distribution sampler.</em>
+<a class="jxr_linenumber" name="L239" href="#L239">239</a> <strong class="jxr_keyword">return</strong> InverseTransformDiscreteSampler.of(rng, <strong class="jxr_keyword">this</strong>::inverseCumulativeProbability)::sample;
+<a class="jxr_linenumber" name="L240" href="#L240">240</a> }
+<a class="jxr_linenumber" name="L241" href="#L241">241</a> }
+</pre>
+<hr/>
+<div id="footer">Copyright © 2018–2022 <a href="https://www.apache.org/">The Apache Software Foundation</a>. All rights reserved.</div>
+</body>
+</html>
Added: dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/ArgumentUtils.html
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--- dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/ArgumentUtils.html (added)
+++ dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/ArgumentUtils.html Thu Dec 1 16:47:12 2022
@@ -0,0 +1,62 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
+<head><meta http-equiv="content-type" content="text/html; charset=UTF-8" />
+<title>ArgumentUtils xref</title>
+<link type="text/css" rel="stylesheet" href="../../../../../stylesheet.css" />
+</head>
+<body>
+<div id="overview"><a href="../../../../../../apidocs/org/apache/commons/statistics/distribution/ArgumentUtils.html">View Javadoc</a></div><pre>
+<a class="jxr_linenumber" name="L1" href="#L1">1</a> <em class="jxr_comment">/*</em>
+<a class="jxr_linenumber" name="L2" href="#L2">2</a> <em class="jxr_comment"> * Licensed to the Apache Software Foundation (ASF) under one or more</em>
+<a class="jxr_linenumber" name="L3" href="#L3">3</a> <em class="jxr_comment"> * contributor license agreements. See the NOTICE file distributed with</em>
+<a class="jxr_linenumber" name="L4" href="#L4">4</a> <em class="jxr_comment"> * this work for additional information regarding copyright ownership.</em>
+<a class="jxr_linenumber" name="L5" href="#L5">5</a> <em class="jxr_comment"> * The ASF licenses this file to You under the Apache License, Version 2.0</em>
+<a class="jxr_linenumber" name="L6" href="#L6">6</a> <em class="jxr_comment"> * (the "License"); you may not use this file except in compliance with</em>
+<a class="jxr_linenumber" name="L7" href="#L7">7</a> <em class="jxr_comment"> * the License. You may obtain a copy of the License at</em>
+<a class="jxr_linenumber" name="L8" href="#L8">8</a> <em class="jxr_comment"> *</em>
+<a class="jxr_linenumber" name="L9" href="#L9">9</a> <em class="jxr_comment"> * <a href="http://www.apache.org/licenses/LICENSE-2.0" target="alexandria_uri">http://www.apache.org/licenses/LICENSE-2.0</a></em>
+<a class="jxr_linenumber" name="L10" href="#L10">10</a> <em class="jxr_comment"> *</em>
+<a class="jxr_linenumber" name="L11" href="#L11">11</a> <em class="jxr_comment"> * Unless required by applicable law or agreed to in writing, software</em>
+<a class="jxr_linenumber" name="L12" href="#L12">12</a> <em class="jxr_comment"> * distributed under the License is distributed on an "AS IS" BASIS,</em>
+<a class="jxr_linenumber" name="L13" href="#L13">13</a> <em class="jxr_comment"> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</em>
+<a class="jxr_linenumber" name="L14" href="#L14">14</a> <em class="jxr_comment"> * See the License for the specific language governing permissions and</em>
+<a class="jxr_linenumber" name="L15" href="#L15">15</a> <em class="jxr_comment"> * limitations under the License.</em>
+<a class="jxr_linenumber" name="L16" href="#L16">16</a> <em class="jxr_comment"> */</em>
+<a class="jxr_linenumber" name="L17" href="#L17">17</a> <strong class="jxr_keyword">package</strong> org.apache.commons.statistics.distribution;
+<a class="jxr_linenumber" name="L18" href="#L18">18</a>
+<a class="jxr_linenumber" name="L19" href="#L19">19</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L20" href="#L20">20</a> <em class="jxr_javadoccomment"> * Utilities for argument validation.</em>
+<a class="jxr_linenumber" name="L21" href="#L21">21</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L22" href="#L22">22</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">class</strong> <a name="ArgumentUtils" href="../../../../../org/apache/commons/statistics/distribution/ArgumentUtils.html#ArgumentUtils">ArgumentUtils</a> {
+<a class="jxr_linenumber" name="L23" href="#L23">23</a> <em class="jxr_javadoccomment">/** No instances. */</em>
+<a class="jxr_linenumber" name="L24" href="#L24">24</a> <strong class="jxr_keyword">private</strong> <a name="ArgumentUtils" href="../../../../../org/apache/commons/statistics/distribution/ArgumentUtils.html#ArgumentUtils">ArgumentUtils</a>() {}
+<a class="jxr_linenumber" name="L25" href="#L25">25</a>
+<a class="jxr_linenumber" name="L26" href="#L26">26</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L27" href="#L27">27</a> <em class="jxr_javadoccomment"> * Checks if the value {@code x} is finite and strictly positive.</em>
+<a class="jxr_linenumber" name="L28" href="#L28">28</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L29" href="#L29">29</a> <em class="jxr_javadoccomment"> * @param x Value</em>
+<a class="jxr_linenumber" name="L30" href="#L30">30</a> <em class="jxr_javadoccomment"> * @return true if {@code x > 0} and is finite</em>
+<a class="jxr_linenumber" name="L31" href="#L31">31</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L32" href="#L32">32</a> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">boolean</strong> isFiniteStrictlyPositive(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L33" href="#L33">33</a> <strong class="jxr_keyword">return</strong> x > 0 && x < Double.POSITIVE_INFINITY;
+<a class="jxr_linenumber" name="L34" href="#L34">34</a> }
+<a class="jxr_linenumber" name="L35" href="#L35">35</a>
+<a class="jxr_linenumber" name="L36" href="#L36">36</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L37" href="#L37">37</a> <em class="jxr_javadoccomment"> * Check the probability {@code p} is in the interval {@code [0, 1]}.</em>
+<a class="jxr_linenumber" name="L38" href="#L38">38</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L39" href="#L39">39</a> <em class="jxr_javadoccomment"> * @param p Probability</em>
+<a class="jxr_linenumber" name="L40" href="#L40">40</a> <em class="jxr_javadoccomment"> * @throws IllegalArgumentException if {@code p < 0} or {@code p > 1}</em>
+<a class="jxr_linenumber" name="L41" href="#L41">41</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L42" href="#L42">42</a> <strong class="jxr_keyword">static</strong> <strong class="jxr_keyword">void</strong> checkProbability(<strong class="jxr_keyword">double</strong> p) {
+<a class="jxr_linenumber" name="L43" href="#L43">43</a> <strong class="jxr_keyword">if</strong> (p >= 0 && p <= 1) {
+<a class="jxr_linenumber" name="L44" href="#L44">44</a> <strong class="jxr_keyword">return</strong>;
+<a class="jxr_linenumber" name="L45" href="#L45">45</a> }
+<a class="jxr_linenumber" name="L46" href="#L46">46</a> <em class="jxr_comment">// Out-of-range or NaN</em>
+<a class="jxr_linenumber" name="L47" href="#L47">47</a> <strong class="jxr_keyword">throw</strong> <strong class="jxr_keyword">new</strong> <a name="DistributionException" href="../../../../../org/apache/commons/statistics/distribution/DistributionException.html#DistributionException">DistributionException</a>(DistributionException.INVALID_PROBABILITY, p);
+<a class="jxr_linenumber" name="L48" href="#L48">48</a> }
+<a class="jxr_linenumber" name="L49" href="#L49">49</a> }
+</pre>
+<hr/>
+<div id="footer">Copyright © 2018–2022 <a href="https://www.apache.org/">The Apache Software Foundation</a>. All rights reserved.</div>
+</body>
+</html>
Added: dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/BetaDistribution.html
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--- dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/BetaDistribution.html (added)
+++ dev/commons/statistics/1.0-RC1/site/xref/org/apache/commons/statistics/distribution/BetaDistribution.html Thu Dec 1 16:47:12 2022
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
+<head><meta http-equiv="content-type" content="text/html; charset=UTF-8" />
+<title>BetaDistribution xref</title>
+<link type="text/css" rel="stylesheet" href="../../../../../stylesheet.css" />
+</head>
+<body>
+<div id="overview"><a href="../../../../../../apidocs/org/apache/commons/statistics/distribution/BetaDistribution.html">View Javadoc</a></div><pre>
+<a class="jxr_linenumber" name="L1" href="#L1">1</a> <em class="jxr_comment">/*</em>
+<a class="jxr_linenumber" name="L2" href="#L2">2</a> <em class="jxr_comment"> * Licensed to the Apache Software Foundation (ASF) under one or more</em>
+<a class="jxr_linenumber" name="L3" href="#L3">3</a> <em class="jxr_comment"> * contributor license agreements. See the NOTICE file distributed with</em>
+<a class="jxr_linenumber" name="L4" href="#L4">4</a> <em class="jxr_comment"> * this work for additional information regarding copyright ownership.</em>
+<a class="jxr_linenumber" name="L5" href="#L5">5</a> <em class="jxr_comment"> * The ASF licenses this file to You under the Apache License, Version 2.0</em>
+<a class="jxr_linenumber" name="L6" href="#L6">6</a> <em class="jxr_comment"> * (the "License"); you may not use this file except in compliance with</em>
+<a class="jxr_linenumber" name="L7" href="#L7">7</a> <em class="jxr_comment"> * the License. You may obtain a copy of the License at</em>
+<a class="jxr_linenumber" name="L8" href="#L8">8</a> <em class="jxr_comment"> *</em>
+<a class="jxr_linenumber" name="L9" href="#L9">9</a> <em class="jxr_comment"> * <a href="http://www.apache.org/licenses/LICENSE-2.0" target="alexandria_uri">http://www.apache.org/licenses/LICENSE-2.0</a></em>
+<a class="jxr_linenumber" name="L10" href="#L10">10</a> <em class="jxr_comment"> *</em>
+<a class="jxr_linenumber" name="L11" href="#L11">11</a> <em class="jxr_comment"> * Unless required by applicable law or agreed to in writing, software</em>
+<a class="jxr_linenumber" name="L12" href="#L12">12</a> <em class="jxr_comment"> * distributed under the License is distributed on an "AS IS" BASIS,</em>
+<a class="jxr_linenumber" name="L13" href="#L13">13</a> <em class="jxr_comment"> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.</em>
+<a class="jxr_linenumber" name="L14" href="#L14">14</a> <em class="jxr_comment"> * See the License for the specific language governing permissions and</em>
+<a class="jxr_linenumber" name="L15" href="#L15">15</a> <em class="jxr_comment"> * limitations under the License.</em>
+<a class="jxr_linenumber" name="L16" href="#L16">16</a> <em class="jxr_comment"> */</em>
+<a class="jxr_linenumber" name="L17" href="#L17">17</a> <strong class="jxr_keyword">package</strong> org.apache.commons.statistics.distribution;
+<a class="jxr_linenumber" name="L18" href="#L18">18</a>
+<a class="jxr_linenumber" name="L19" href="#L19">19</a> <strong class="jxr_keyword">import</strong> org.apache.commons.numbers.gamma.RegularizedBeta;
+<a class="jxr_linenumber" name="L20" href="#L20">20</a> <strong class="jxr_keyword">import</strong> org.apache.commons.numbers.gamma.LogBeta;
+<a class="jxr_linenumber" name="L21" href="#L21">21</a> <strong class="jxr_keyword">import</strong> org.apache.commons.rng.UniformRandomProvider;
+<a class="jxr_linenumber" name="L22" href="#L22">22</a> <strong class="jxr_keyword">import</strong> org.apache.commons.rng.sampling.distribution.ChengBetaSampler;
+<a class="jxr_linenumber" name="L23" href="#L23">23</a>
+<a class="jxr_linenumber" name="L24" href="#L24">24</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L25" href="#L25">25</a> <em class="jxr_javadoccomment"> * Implementation of the beta distribution.</em>
+<a class="jxr_linenumber" name="L26" href="#L26">26</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L27" href="#L27">27</a> <em class="jxr_javadoccomment"> * <p>The probability density function of \( X \) is:</em>
+<a class="jxr_linenumber" name="L28" href="#L28">28</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L29" href="#L29">29</a> <em class="jxr_javadoccomment"> * <p>\[ f(x; \alpha, \beta) = \frac{1}{ B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta-1} \]</em>
+<a class="jxr_linenumber" name="L30" href="#L30">30</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L31" href="#L31">31</a> <em class="jxr_javadoccomment"> * <p>for \( \alpha &gt; 0 \),</em>
+<a class="jxr_linenumber" name="L32" href="#L32">32</a> <em class="jxr_javadoccomment"> * \( \beta &gt; 0 \), \( x \in [0, 1] \), and</em>
+<a class="jxr_linenumber" name="L33" href="#L33">33</a> <em class="jxr_javadoccomment"> * the beta function, \( B \), is a normalization constant:</em>
+<a class="jxr_linenumber" name="L34" href="#L34">34</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L35" href="#L35">35</a> <em class="jxr_javadoccomment"> * <p>\[ B(\alpha, \beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} \]</em>
+<a class="jxr_linenumber" name="L36" href="#L36">36</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L37" href="#L37">37</a> <em class="jxr_javadoccomment"> * <p>where \( \Gamma \) is the gamma function.</em>
+<a class="jxr_linenumber" name="L38" href="#L38">38</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L39" href="#L39">39</a> <em class="jxr_javadoccomment"> * <p>\( \alpha \) and \( \beta \) are <em>shape</em> parameters.</em>
+<a class="jxr_linenumber" name="L40" href="#L40">40</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L41" href="#L41">41</a> <em class="jxr_javadoccomment"> * @see <a href="<a href="https://en.wikipedia.org/wiki/Beta_distribution" target="alexandria_uri">https://en.wikipedia.org/wiki/Beta_distribution</a>">Beta distribution (Wikipedia)</a></em>
+<a class="jxr_linenumber" name="L42" href="#L42">42</a> <em class="jxr_javadoccomment"> * @see <a href="<a href="https://mathworld.wolfram.com/BetaDistribution.html" target="alexandria_uri">https://mathworld.wolfram.com/BetaDistribution.html</a>">Beta distribution (MathWorld)</a></em>
+<a class="jxr_linenumber" name="L43" href="#L43">43</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L44" href="#L44">44</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">class</strong> <a name="BetaDistribution" href="../../../../../org/apache/commons/statistics/distribution/BetaDistribution.html#BetaDistribution">BetaDistribution</a> <strong class="jxr_keyword">extends</strong> <a name="AbstractContinuousDistribution" href="../../../../../org/apache/commons/statistics/distribution/AbstractContinuousDistribution.html#AbstractContinuousDistribution">AbstractContinuousDistribution</a> {
+<a class="jxr_linenumber" name="L45" href="#L45">45</a> <em class="jxr_javadoccomment">/** First shape parameter. */</em>
+<a class="jxr_linenumber" name="L46" href="#L46">46</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> alpha;
+<a class="jxr_linenumber" name="L47" href="#L47">47</a> <em class="jxr_javadoccomment">/** Second shape parameter. */</em>
+<a class="jxr_linenumber" name="L48" href="#L48">48</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> beta;
+<a class="jxr_linenumber" name="L49" href="#L49">49</a> <em class="jxr_javadoccomment">/** Normalizing factor used in log density computations. log(beta(a, b)). */</em>
+<a class="jxr_linenumber" name="L50" href="#L50">50</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> logBeta;
+<a class="jxr_linenumber" name="L51" href="#L51">51</a> <em class="jxr_javadoccomment">/** Cached value for inverse probability function. */</em>
+<a class="jxr_linenumber" name="L52" href="#L52">52</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> mean;
+<a class="jxr_linenumber" name="L53" href="#L53">53</a> <em class="jxr_javadoccomment">/** Cached value for inverse probability function. */</em>
+<a class="jxr_linenumber" name="L54" href="#L54">54</a> <strong class="jxr_keyword">private</strong> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> variance;
+<a class="jxr_linenumber" name="L55" href="#L55">55</a>
+<a class="jxr_linenumber" name="L56" href="#L56">56</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L57" href="#L57">57</a> <em class="jxr_javadoccomment"> * @param alpha First shape parameter (must be positive).</em>
+<a class="jxr_linenumber" name="L58" href="#L58">58</a> <em class="jxr_javadoccomment"> * @param beta Second shape parameter (must be positive).</em>
+<a class="jxr_linenumber" name="L59" href="#L59">59</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L60" href="#L60">60</a> <strong class="jxr_keyword">private</strong> <a name="BetaDistribution" href="../../../../../org/apache/commons/statistics/distribution/BetaDistribution.html#BetaDistribution">BetaDistribution</a>(<strong class="jxr_keyword">double</strong> alpha,
+<a class="jxr_linenumber" name="L61" href="#L61">61</a> <strong class="jxr_keyword">double</strong> beta) {
+<a class="jxr_linenumber" name="L62" href="#L62">62</a> <strong class="jxr_keyword">this</strong>.alpha = alpha;
+<a class="jxr_linenumber" name="L63" href="#L63">63</a> <strong class="jxr_keyword">this</strong>.beta = beta;
+<a class="jxr_linenumber" name="L64" href="#L64">64</a> logBeta = LogBeta.value(alpha, beta);
+<a class="jxr_linenumber" name="L65" href="#L65">65</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> alphabetasum = alpha + beta;
+<a class="jxr_linenumber" name="L66" href="#L66">66</a> mean = alpha / alphabetasum;
+<a class="jxr_linenumber" name="L67" href="#L67">67</a> variance = (alpha * beta) / ((alphabetasum * alphabetasum) * (alphabetasum + 1));
+<a class="jxr_linenumber" name="L68" href="#L68">68</a> }
+<a class="jxr_linenumber" name="L69" href="#L69">69</a>
+<a class="jxr_linenumber" name="L70" href="#L70">70</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L71" href="#L71">71</a> <em class="jxr_javadoccomment"> * Creates a beta distribution.</em>
+<a class="jxr_linenumber" name="L72" href="#L72">72</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L73" href="#L73">73</a> <em class="jxr_javadoccomment"> * @param alpha First shape parameter (must be positive).</em>
+<a class="jxr_linenumber" name="L74" href="#L74">74</a> <em class="jxr_javadoccomment"> * @param beta Second shape parameter (must be positive).</em>
+<a class="jxr_linenumber" name="L75" href="#L75">75</a> <em class="jxr_javadoccomment"> * @return the distribution</em>
+<a class="jxr_linenumber" name="L76" href="#L76">76</a> <em class="jxr_javadoccomment"> * @throws IllegalArgumentException if {@code alpha <= 0} or {@code beta <= 0}.</em>
+<a class="jxr_linenumber" name="L77" href="#L77">77</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L78" href="#L78">78</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">static</strong> <a name="BetaDistribution" href="../../../../../org/apache/commons/statistics/distribution/BetaDistribution.html#BetaDistribution">BetaDistribution</a> of(<strong class="jxr_keyword">double</strong> alpha,
+<a class="jxr_linenumber" name="L79" href="#L79">79</a> <strong class="jxr_keyword">double</strong> beta) {
+<a class="jxr_linenumber" name="L80" href="#L80">80</a> <strong class="jxr_keyword">if</strong> (alpha <= 0) {
+<a class="jxr_linenumber" name="L81" href="#L81">81</a> <strong class="jxr_keyword">throw</strong> <strong class="jxr_keyword">new</strong> <a name="DistributionException" href="../../../../../org/apache/commons/statistics/distribution/DistributionException.html#DistributionException">DistributionException</a>(DistributionException.NOT_STRICTLY_POSITIVE, alpha);
+<a class="jxr_linenumber" name="L82" href="#L82">82</a> }
+<a class="jxr_linenumber" name="L83" href="#L83">83</a> <strong class="jxr_keyword">if</strong> (beta <= 0) {
+<a class="jxr_linenumber" name="L84" href="#L84">84</a> <strong class="jxr_keyword">throw</strong> <strong class="jxr_keyword">new</strong> <a name="DistributionException" href="../../../../../org/apache/commons/statistics/distribution/DistributionException.html#DistributionException">DistributionException</a>(DistributionException.NOT_STRICTLY_POSITIVE, beta);
+<a class="jxr_linenumber" name="L85" href="#L85">85</a> }
+<a class="jxr_linenumber" name="L86" href="#L86">86</a> <strong class="jxr_keyword">return</strong> <strong class="jxr_keyword">new</strong> <a name="BetaDistribution" href="../../../../../org/apache/commons/statistics/distribution/BetaDistribution.html#BetaDistribution">BetaDistribution</a>(alpha, beta);
+<a class="jxr_linenumber" name="L87" href="#L87">87</a> }
+<a class="jxr_linenumber" name="L88" href="#L88">88</a>
+<a class="jxr_linenumber" name="L89" href="#L89">89</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L90" href="#L90">90</a> <em class="jxr_javadoccomment"> * Gets the first shape parameter of this distribution.</em>
+<a class="jxr_linenumber" name="L91" href="#L91">91</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L92" href="#L92">92</a> <em class="jxr_javadoccomment"> * @return the first shape parameter.</em>
+<a class="jxr_linenumber" name="L93" href="#L93">93</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L94" href="#L94">94</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getAlpha() {
+<a class="jxr_linenumber" name="L95" href="#L95">95</a> <strong class="jxr_keyword">return</strong> alpha;
+<a class="jxr_linenumber" name="L96" href="#L96">96</a> }
+<a class="jxr_linenumber" name="L97" href="#L97">97</a>
+<a class="jxr_linenumber" name="L98" href="#L98">98</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L99" href="#L99">99</a> <em class="jxr_javadoccomment"> * Gets the second shape parameter of this distribution.</em>
+<a class="jxr_linenumber" name="L100" href="#L100">100</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L101" href="#L101">101</a> <em class="jxr_javadoccomment"> * @return the second shape parameter.</em>
+<a class="jxr_linenumber" name="L102" href="#L102">102</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L103" href="#L103">103</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getBeta() {
+<a class="jxr_linenumber" name="L104" href="#L104">104</a> <strong class="jxr_keyword">return</strong> beta;
+<a class="jxr_linenumber" name="L105" href="#L105">105</a> }
+<a class="jxr_linenumber" name="L106" href="#L106">106</a>
+<a class="jxr_linenumber" name="L107" href="#L107">107</a> <em class="jxr_javadoccomment">/** {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L108" href="#L108">108</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L109" href="#L109">109</a> <em class="jxr_javadoccomment"> * <p>The density is not defined when {@code x = 0, alpha < 1}, or {@code x = 1, beta < 1}.</em>
+<a class="jxr_linenumber" name="L110" href="#L110">110</a> <em class="jxr_javadoccomment"> * In this case the limit of infinity is returned.</em>
+<a class="jxr_linenumber" name="L111" href="#L111">111</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L112" href="#L112">112</a> @Override
+<a class="jxr_linenumber" name="L113" href="#L113">113</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> density(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L114" href="#L114">114</a> <strong class="jxr_keyword">if</strong> (x < 0 || x > 1) {
+<a class="jxr_linenumber" name="L115" href="#L115">115</a> <strong class="jxr_keyword">return</strong> 0;
+<a class="jxr_linenumber" name="L116" href="#L116">116</a> }
+<a class="jxr_linenumber" name="L117" href="#L117">117</a> <strong class="jxr_keyword">return</strong> RegularizedBeta.derivative(x, alpha, beta);
+<a class="jxr_linenumber" name="L118" href="#L118">118</a> }
+<a class="jxr_linenumber" name="L119" href="#L119">119</a>
+<a class="jxr_linenumber" name="L120" href="#L120">120</a> <em class="jxr_javadoccomment">/** {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L121" href="#L121">121</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L122" href="#L122">122</a> <em class="jxr_javadoccomment"> * <p>The density is not defined when {@code x = 0, alpha < 1}, or {@code x = 1, beta < 1}.</em>
+<a class="jxr_linenumber" name="L123" href="#L123">123</a> <em class="jxr_javadoccomment"> * In this case the limit of infinity is returned.</em>
+<a class="jxr_linenumber" name="L124" href="#L124">124</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L125" href="#L125">125</a> @Override
+<a class="jxr_linenumber" name="L126" href="#L126">126</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> logDensity(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L127" href="#L127">127</a> <strong class="jxr_keyword">if</strong> (x < 0 || x > 1) {
+<a class="jxr_linenumber" name="L128" href="#L128">128</a> <strong class="jxr_keyword">return</strong> Double.NEGATIVE_INFINITY;
+<a class="jxr_linenumber" name="L129" href="#L129">129</a> } <strong class="jxr_keyword">else</strong> <strong class="jxr_keyword">if</strong> (x == 0) {
+<a class="jxr_linenumber" name="L130" href="#L130">130</a> <strong class="jxr_keyword">if</strong> (alpha < 1) {
+<a class="jxr_linenumber" name="L131" href="#L131">131</a> <em class="jxr_comment">// Distribution is not valid when x=0, alpha<1</em>
+<a class="jxr_linenumber" name="L132" href="#L132">132</a> <em class="jxr_comment">// due to a divide by zero error.</em>
+<a class="jxr_linenumber" name="L133" href="#L133">133</a> <em class="jxr_comment">// Do not raise an exception and return the limit.</em>
+<a class="jxr_linenumber" name="L134" href="#L134">134</a> <strong class="jxr_keyword">return</strong> Double.POSITIVE_INFINITY;
+<a class="jxr_linenumber" name="L135" href="#L135">135</a> }
+<a class="jxr_linenumber" name="L136" href="#L136">136</a> <em class="jxr_comment">// Special case of cancellation: x^(a-1) (1-x)^(b-1) / B(a, b) = 1 / B(a, b)</em>
+<a class="jxr_linenumber" name="L137" href="#L137">137</a> <strong class="jxr_keyword">if</strong> (alpha == 1) {
+<a class="jxr_linenumber" name="L138" href="#L138">138</a> <strong class="jxr_keyword">return</strong> -logBeta;
+<a class="jxr_linenumber" name="L139" href="#L139">139</a> }
+<a class="jxr_linenumber" name="L140" href="#L140">140</a> <strong class="jxr_keyword">return</strong> Double.NEGATIVE_INFINITY;
+<a class="jxr_linenumber" name="L141" href="#L141">141</a> } <strong class="jxr_keyword">else</strong> <strong class="jxr_keyword">if</strong> (x == 1) {
+<a class="jxr_linenumber" name="L142" href="#L142">142</a> <strong class="jxr_keyword">if</strong> (beta < 1) {
+<a class="jxr_linenumber" name="L143" href="#L143">143</a> <em class="jxr_comment">// Distribution is not valid when x=1, beta<1</em>
+<a class="jxr_linenumber" name="L144" href="#L144">144</a> <em class="jxr_comment">// due to a divide by zero error.</em>
+<a class="jxr_linenumber" name="L145" href="#L145">145</a> <em class="jxr_comment">// Do not raise an exception and return the limit.</em>
+<a class="jxr_linenumber" name="L146" href="#L146">146</a> <strong class="jxr_keyword">return</strong> Double.POSITIVE_INFINITY;
+<a class="jxr_linenumber" name="L147" href="#L147">147</a> }
+<a class="jxr_linenumber" name="L148" href="#L148">148</a> <em class="jxr_comment">// Special case of cancellation: x^(a-1) (1-x)^(b-1) / B(a, b) = 1 / B(a, b)</em>
+<a class="jxr_linenumber" name="L149" href="#L149">149</a> <strong class="jxr_keyword">if</strong> (beta == 1) {
+<a class="jxr_linenumber" name="L150" href="#L150">150</a> <strong class="jxr_keyword">return</strong> -logBeta;
+<a class="jxr_linenumber" name="L151" href="#L151">151</a> }
+<a class="jxr_linenumber" name="L152" href="#L152">152</a> <strong class="jxr_keyword">return</strong> Double.NEGATIVE_INFINITY;
+<a class="jxr_linenumber" name="L153" href="#L153">153</a> }
+<a class="jxr_linenumber" name="L154" href="#L154">154</a>
+<a class="jxr_linenumber" name="L155" href="#L155">155</a> <em class="jxr_comment">// Log computation</em>
+<a class="jxr_linenumber" name="L156" href="#L156">156</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> logX = Math.log(x);
+<a class="jxr_linenumber" name="L157" href="#L157">157</a> <strong class="jxr_keyword">final</strong> <strong class="jxr_keyword">double</strong> log1mX = Math.log1p(-x);
+<a class="jxr_linenumber" name="L158" href="#L158">158</a> <strong class="jxr_keyword">return</strong> (alpha - 1) * logX + (beta - 1) * log1mX - logBeta;
+<a class="jxr_linenumber" name="L159" href="#L159">159</a> }
+<a class="jxr_linenumber" name="L160" href="#L160">160</a>
+<a class="jxr_linenumber" name="L161" href="#L161">161</a> <em class="jxr_javadoccomment">/** {@inheritDoc} */</em>
+<a class="jxr_linenumber" name="L162" href="#L162">162</a> @Override
+<a class="jxr_linenumber" name="L163" href="#L163">163</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> cumulativeProbability(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L164" href="#L164">164</a> <strong class="jxr_keyword">if</strong> (x <= 0) {
+<a class="jxr_linenumber" name="L165" href="#L165">165</a> <strong class="jxr_keyword">return</strong> 0;
+<a class="jxr_linenumber" name="L166" href="#L166">166</a> } <strong class="jxr_keyword">else</strong> <strong class="jxr_keyword">if</strong> (x >= 1) {
+<a class="jxr_linenumber" name="L167" href="#L167">167</a> <strong class="jxr_keyword">return</strong> 1;
+<a class="jxr_linenumber" name="L168" href="#L168">168</a> } <strong class="jxr_keyword">else</strong> {
+<a class="jxr_linenumber" name="L169" href="#L169">169</a> <strong class="jxr_keyword">return</strong> RegularizedBeta.value(x, alpha, beta);
+<a class="jxr_linenumber" name="L170" href="#L170">170</a> }
+<a class="jxr_linenumber" name="L171" href="#L171">171</a> }
+<a class="jxr_linenumber" name="L172" href="#L172">172</a>
+<a class="jxr_linenumber" name="L173" href="#L173">173</a> <em class="jxr_javadoccomment">/** {@inheritDoc} */</em>
+<a class="jxr_linenumber" name="L174" href="#L174">174</a> @Override
+<a class="jxr_linenumber" name="L175" href="#L175">175</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> survivalProbability(<strong class="jxr_keyword">double</strong> x) {
+<a class="jxr_linenumber" name="L176" href="#L176">176</a> <strong class="jxr_keyword">if</strong> (x <= 0) {
+<a class="jxr_linenumber" name="L177" href="#L177">177</a> <strong class="jxr_keyword">return</strong> 1;
+<a class="jxr_linenumber" name="L178" href="#L178">178</a> } <strong class="jxr_keyword">else</strong> <strong class="jxr_keyword">if</strong> (x >= 1) {
+<a class="jxr_linenumber" name="L179" href="#L179">179</a> <strong class="jxr_keyword">return</strong> 0;
+<a class="jxr_linenumber" name="L180" href="#L180">180</a> } <strong class="jxr_keyword">else</strong> {
+<a class="jxr_linenumber" name="L181" href="#L181">181</a> <strong class="jxr_keyword">return</strong> RegularizedBeta.complement(x, alpha, beta);
+<a class="jxr_linenumber" name="L182" href="#L182">182</a> }
+<a class="jxr_linenumber" name="L183" href="#L183">183</a> }
+<a class="jxr_linenumber" name="L184" href="#L184">184</a>
+<a class="jxr_linenumber" name="L185" href="#L185">185</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L186" href="#L186">186</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L187" href="#L187">187</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L188" href="#L188">188</a> <em class="jxr_javadoccomment"> * <p>For first shape parameter \( \alpha \) and second shape parameter</em>
+<a class="jxr_linenumber" name="L189" href="#L189">189</a> <em class="jxr_javadoccomment"> * \( \beta \), the mean is:</em>
+<a class="jxr_linenumber" name="L190" href="#L190">190</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L191" href="#L191">191</a> <em class="jxr_javadoccomment"> * <p>\[ \frac{\alpha}{\alpha + \beta} \]</em>
+<a class="jxr_linenumber" name="L192" href="#L192">192</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L193" href="#L193">193</a> @Override
+<a class="jxr_linenumber" name="L194" href="#L194">194</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getMean() {
+<a class="jxr_linenumber" name="L195" href="#L195">195</a> <strong class="jxr_keyword">return</strong> mean;
+<a class="jxr_linenumber" name="L196" href="#L196">196</a> }
+<a class="jxr_linenumber" name="L197" href="#L197">197</a>
+<a class="jxr_linenumber" name="L198" href="#L198">198</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L199" href="#L199">199</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L200" href="#L200">200</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L201" href="#L201">201</a> <em class="jxr_javadoccomment"> * <p>For first shape parameter \( \alpha \) and second shape parameter</em>
+<a class="jxr_linenumber" name="L202" href="#L202">202</a> <em class="jxr_javadoccomment"> * \( \beta \), the variance is:</em>
+<a class="jxr_linenumber" name="L203" href="#L203">203</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L204" href="#L204">204</a> <em class="jxr_javadoccomment"> * <p>\[ \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \].</em>
+<a class="jxr_linenumber" name="L205" href="#L205">205</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L206" href="#L206">206</a> @Override
+<a class="jxr_linenumber" name="L207" href="#L207">207</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getVariance() {
+<a class="jxr_linenumber" name="L208" href="#L208">208</a> <strong class="jxr_keyword">return</strong> variance;
+<a class="jxr_linenumber" name="L209" href="#L209">209</a> }
+<a class="jxr_linenumber" name="L210" href="#L210">210</a>
+<a class="jxr_linenumber" name="L211" href="#L211">211</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L212" href="#L212">212</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L213" href="#L213">213</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L214" href="#L214">214</a> <em class="jxr_javadoccomment"> * <p>The lower bound of the support is always 0.</em>
+<a class="jxr_linenumber" name="L215" href="#L215">215</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L216" href="#L216">216</a> <em class="jxr_javadoccomment"> * @return 0.</em>
+<a class="jxr_linenumber" name="L217" href="#L217">217</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L218" href="#L218">218</a> @Override
+<a class="jxr_linenumber" name="L219" href="#L219">219</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getSupportLowerBound() {
+<a class="jxr_linenumber" name="L220" href="#L220">220</a> <strong class="jxr_keyword">return</strong> 0;
+<a class="jxr_linenumber" name="L221" href="#L221">221</a> }
+<a class="jxr_linenumber" name="L222" href="#L222">222</a>
+<a class="jxr_linenumber" name="L223" href="#L223">223</a> <em class="jxr_javadoccomment">/**</em>
+<a class="jxr_linenumber" name="L224" href="#L224">224</a> <em class="jxr_javadoccomment"> * {@inheritDoc}</em>
+<a class="jxr_linenumber" name="L225" href="#L225">225</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L226" href="#L226">226</a> <em class="jxr_javadoccomment"> * <p>The upper bound of the support is always 1.</em>
+<a class="jxr_linenumber" name="L227" href="#L227">227</a> <em class="jxr_javadoccomment"> *</em>
+<a class="jxr_linenumber" name="L228" href="#L228">228</a> <em class="jxr_javadoccomment"> * @return 1.</em>
+<a class="jxr_linenumber" name="L229" href="#L229">229</a> <em class="jxr_javadoccomment"> */</em>
+<a class="jxr_linenumber" name="L230" href="#L230">230</a> @Override
+<a class="jxr_linenumber" name="L231" href="#L231">231</a> <strong class="jxr_keyword">public</strong> <strong class="jxr_keyword">double</strong> getSupportUpperBound() {
+<a class="jxr_linenumber" name="L232" href="#L232">232</a> <strong class="jxr_keyword">return</strong> 1;
+<a class="jxr_linenumber" name="L233" href="#L233">233</a> }
+<a class="jxr_linenumber" name="L234" href="#L234">234</a>
+<a class="jxr_linenumber" name="L235" href="#L235">235</a> <em class="jxr_javadoccomment">/** {@inheritDoc} */</em>
+<a class="jxr_linenumber" name="L236" href="#L236">236</a> @Override
+<a class="jxr_linenumber" name="L237" href="#L237">237</a> <strong class="jxr_keyword">public</strong> ContinuousDistribution.Sampler createSampler(<strong class="jxr_keyword">final</strong> UniformRandomProvider rng) {
+<a class="jxr_linenumber" name="L238" href="#L238">238</a> <em class="jxr_comment">// Beta distribution sampler.</em>
+<a class="jxr_linenumber" name="L239" href="#L239">239</a> <strong class="jxr_keyword">return</strong> ChengBetaSampler.of(rng, alpha, beta)::sample;
+<a class="jxr_linenumber" name="L240" href="#L240">240</a> }
+<a class="jxr_linenumber" name="L241" href="#L241">241</a> }
+</pre>
+<hr/>
+<div id="footer">Copyright © 2018–2022 <a href="https://www.apache.org/">The Apache Software Foundation</a>. All rights reserved.</div>
+</body>
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