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comparison libs/commons-math-2.1/docs/apidocs/src-html/org/apache/commons/math/distribution/AbstractIntegerDistribution.html @ 13:cbf34dd4d7e6

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commons-math-2.1 added
author dwinter
date Tue, 04 Jan 2011 10:02:07 +0100
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12:970d26a94fb7 13:cbf34dd4d7e6
1 <HTML>
2 <BODY BGCOLOR="white">
3 <PRE>
4 <FONT color="green">001</FONT> /*<a name="line.1"></a>
5 <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
6 <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a>
7 <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a>
8 <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
9 <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
10 <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a>
11 <FONT color="green">008</FONT> *<a name="line.8"></a>
12 <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
13 <FONT color="green">010</FONT> *<a name="line.10"></a>
14 <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
15 <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
16 <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
17 <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a>
18 <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a>
19 <FONT color="green">016</FONT> */<a name="line.16"></a>
20 <FONT color="green">017</FONT> package org.apache.commons.math.distribution;<a name="line.17"></a>
21 <FONT color="green">018</FONT> <a name="line.18"></a>
22 <FONT color="green">019</FONT> import java.io.Serializable;<a name="line.19"></a>
23 <FONT color="green">020</FONT> <a name="line.20"></a>
24 <FONT color="green">021</FONT> import org.apache.commons.math.FunctionEvaluationException;<a name="line.21"></a>
25 <FONT color="green">022</FONT> import org.apache.commons.math.MathException;<a name="line.22"></a>
26 <FONT color="green">023</FONT> import org.apache.commons.math.MathRuntimeException;<a name="line.23"></a>
27 <FONT color="green">024</FONT> <a name="line.24"></a>
28 <FONT color="green">025</FONT> <a name="line.25"></a>
29 <FONT color="green">026</FONT> /**<a name="line.26"></a>
30 <FONT color="green">027</FONT> * Base class for integer-valued discrete distributions. Default<a name="line.27"></a>
31 <FONT color="green">028</FONT> * implementations are provided for some of the methods that do not vary<a name="line.28"></a>
32 <FONT color="green">029</FONT> * from distribution to distribution.<a name="line.29"></a>
33 <FONT color="green">030</FONT> *<a name="line.30"></a>
34 <FONT color="green">031</FONT> * @version $Revision: 920558 $ $Date: 2010-03-08 17:57:32 -0500 (Mon, 08 Mar 2010) $<a name="line.31"></a>
35 <FONT color="green">032</FONT> */<a name="line.32"></a>
36 <FONT color="green">033</FONT> public abstract class AbstractIntegerDistribution extends AbstractDistribution<a name="line.33"></a>
37 <FONT color="green">034</FONT> implements IntegerDistribution, Serializable {<a name="line.34"></a>
38 <FONT color="green">035</FONT> <a name="line.35"></a>
39 <FONT color="green">036</FONT> /** Message for endpoints in wrong order. */<a name="line.36"></a>
40 <FONT color="green">037</FONT> private static final String WRONG_ORDER_ENDPOINTS_MESSAGE =<a name="line.37"></a>
41 <FONT color="green">038</FONT> "lower endpoint ({0}) must be less than or equal to upper endpoint ({1})";<a name="line.38"></a>
42 <FONT color="green">039</FONT> <a name="line.39"></a>
43 <FONT color="green">040</FONT> /** Message for out of range point. */<a name="line.40"></a>
44 <FONT color="green">041</FONT> private static final String OUT_OF_RANGE_POINT =<a name="line.41"></a>
45 <FONT color="green">042</FONT> "{0} out of [{1}, {2}] range";<a name="line.42"></a>
46 <FONT color="green">043</FONT> <a name="line.43"></a>
47 <FONT color="green">044</FONT> /** Serializable version identifier */<a name="line.44"></a>
48 <FONT color="green">045</FONT> private static final long serialVersionUID = -1146319659338487221L;<a name="line.45"></a>
49 <FONT color="green">046</FONT> <a name="line.46"></a>
50 <FONT color="green">047</FONT> /**<a name="line.47"></a>
51 <FONT color="green">048</FONT> * Default constructor.<a name="line.48"></a>
52 <FONT color="green">049</FONT> */<a name="line.49"></a>
53 <FONT color="green">050</FONT> protected AbstractIntegerDistribution() {<a name="line.50"></a>
54 <FONT color="green">051</FONT> super();<a name="line.51"></a>
55 <FONT color="green">052</FONT> }<a name="line.52"></a>
56 <FONT color="green">053</FONT> <a name="line.53"></a>
57 <FONT color="green">054</FONT> /**<a name="line.54"></a>
58 <FONT color="green">055</FONT> * For a random variable X whose values are distributed according<a name="line.55"></a>
59 <FONT color="green">056</FONT> * to this distribution, this method returns P(X &amp;le; x). In other words,<a name="line.56"></a>
60 <FONT color="green">057</FONT> * this method represents the (cumulative) distribution function, or<a name="line.57"></a>
61 <FONT color="green">058</FONT> * CDF, for this distribution.<a name="line.58"></a>
62 <FONT color="green">059</FONT> * &lt;p&gt;<a name="line.59"></a>
63 <FONT color="green">060</FONT> * If &lt;code&gt;x&lt;/code&gt; does not represent an integer value, the CDF is<a name="line.60"></a>
64 <FONT color="green">061</FONT> * evaluated at the greatest integer less than x.<a name="line.61"></a>
65 <FONT color="green">062</FONT> *<a name="line.62"></a>
66 <FONT color="green">063</FONT> * @param x the value at which the distribution function is evaluated.<a name="line.63"></a>
67 <FONT color="green">064</FONT> * @return cumulative probability that a random variable with this<a name="line.64"></a>
68 <FONT color="green">065</FONT> * distribution takes a value less than or equal to &lt;code&gt;x&lt;/code&gt;<a name="line.65"></a>
69 <FONT color="green">066</FONT> * @throws MathException if the cumulative probability can not be<a name="line.66"></a>
70 <FONT color="green">067</FONT> * computed due to convergence or other numerical errors.<a name="line.67"></a>
71 <FONT color="green">068</FONT> */<a name="line.68"></a>
72 <FONT color="green">069</FONT> public double cumulativeProbability(double x) throws MathException {<a name="line.69"></a>
73 <FONT color="green">070</FONT> return cumulativeProbability((int) Math.floor(x));<a name="line.70"></a>
74 <FONT color="green">071</FONT> }<a name="line.71"></a>
75 <FONT color="green">072</FONT> <a name="line.72"></a>
76 <FONT color="green">073</FONT> /**<a name="line.73"></a>
77 <FONT color="green">074</FONT> * For a random variable X whose values are distributed according<a name="line.74"></a>
78 <FONT color="green">075</FONT> * to this distribution, this method returns P(x0 &amp;le; X &amp;le; x1).<a name="line.75"></a>
79 <FONT color="green">076</FONT> *<a name="line.76"></a>
80 <FONT color="green">077</FONT> * @param x0 the (inclusive) lower bound<a name="line.77"></a>
81 <FONT color="green">078</FONT> * @param x1 the (inclusive) upper bound<a name="line.78"></a>
82 <FONT color="green">079</FONT> * @return the probability that a random variable with this distribution<a name="line.79"></a>
83 <FONT color="green">080</FONT> * will take a value between &lt;code&gt;x0&lt;/code&gt; and &lt;code&gt;x1&lt;/code&gt;,<a name="line.80"></a>
84 <FONT color="green">081</FONT> * including the endpoints.<a name="line.81"></a>
85 <FONT color="green">082</FONT> * @throws MathException if the cumulative probability can not be<a name="line.82"></a>
86 <FONT color="green">083</FONT> * computed due to convergence or other numerical errors.<a name="line.83"></a>
87 <FONT color="green">084</FONT> * @throws IllegalArgumentException if &lt;code&gt;x0 &gt; x1&lt;/code&gt;<a name="line.84"></a>
88 <FONT color="green">085</FONT> */<a name="line.85"></a>
89 <FONT color="green">086</FONT> @Override<a name="line.86"></a>
90 <FONT color="green">087</FONT> public double cumulativeProbability(double x0, double x1)<a name="line.87"></a>
91 <FONT color="green">088</FONT> throws MathException {<a name="line.88"></a>
92 <FONT color="green">089</FONT> if (x0 &gt; x1) {<a name="line.89"></a>
93 <FONT color="green">090</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.90"></a>
94 <FONT color="green">091</FONT> WRONG_ORDER_ENDPOINTS_MESSAGE, x0, x1);<a name="line.91"></a>
95 <FONT color="green">092</FONT> }<a name="line.92"></a>
96 <FONT color="green">093</FONT> if (Math.floor(x0) &lt; x0) {<a name="line.93"></a>
97 <FONT color="green">094</FONT> return cumulativeProbability(((int) Math.floor(x0)) + 1,<a name="line.94"></a>
98 <FONT color="green">095</FONT> (int) Math.floor(x1)); // don't want to count mass below x0<a name="line.95"></a>
99 <FONT color="green">096</FONT> } else { // x0 is mathematical integer, so use as is<a name="line.96"></a>
100 <FONT color="green">097</FONT> return cumulativeProbability((int) Math.floor(x0),<a name="line.97"></a>
101 <FONT color="green">098</FONT> (int) Math.floor(x1));<a name="line.98"></a>
102 <FONT color="green">099</FONT> }<a name="line.99"></a>
103 <FONT color="green">100</FONT> }<a name="line.100"></a>
104 <FONT color="green">101</FONT> <a name="line.101"></a>
105 <FONT color="green">102</FONT> /**<a name="line.102"></a>
106 <FONT color="green">103</FONT> * For a random variable X whose values are distributed according<a name="line.103"></a>
107 <FONT color="green">104</FONT> * to this distribution, this method returns P(X &amp;le; x). In other words,<a name="line.104"></a>
108 <FONT color="green">105</FONT> * this method represents the probability distribution function, or PDF,<a name="line.105"></a>
109 <FONT color="green">106</FONT> * for this distribution.<a name="line.106"></a>
110 <FONT color="green">107</FONT> *<a name="line.107"></a>
111 <FONT color="green">108</FONT> * @param x the value at which the PDF is evaluated.<a name="line.108"></a>
112 <FONT color="green">109</FONT> * @return PDF for this distribution.<a name="line.109"></a>
113 <FONT color="green">110</FONT> * @throws MathException if the cumulative probability can not be<a name="line.110"></a>
114 <FONT color="green">111</FONT> * computed due to convergence or other numerical errors.<a name="line.111"></a>
115 <FONT color="green">112</FONT> */<a name="line.112"></a>
116 <FONT color="green">113</FONT> public abstract double cumulativeProbability(int x) throws MathException;<a name="line.113"></a>
117 <FONT color="green">114</FONT> <a name="line.114"></a>
118 <FONT color="green">115</FONT> /**<a name="line.115"></a>
119 <FONT color="green">116</FONT> * For a random variable X whose values are distributed according<a name="line.116"></a>
120 <FONT color="green">117</FONT> * to this distribution, this method returns P(X = x). In other words, this<a name="line.117"></a>
121 <FONT color="green">118</FONT> * method represents the probability mass function, or PMF, for the distribution.<a name="line.118"></a>
122 <FONT color="green">119</FONT> * &lt;p&gt;<a name="line.119"></a>
123 <FONT color="green">120</FONT> * If &lt;code&gt;x&lt;/code&gt; does not represent an integer value, 0 is returned.<a name="line.120"></a>
124 <FONT color="green">121</FONT> *<a name="line.121"></a>
125 <FONT color="green">122</FONT> * @param x the value at which the probability density function is evaluated<a name="line.122"></a>
126 <FONT color="green">123</FONT> * @return the value of the probability density function at x<a name="line.123"></a>
127 <FONT color="green">124</FONT> */<a name="line.124"></a>
128 <FONT color="green">125</FONT> public double probability(double x) {<a name="line.125"></a>
129 <FONT color="green">126</FONT> double fl = Math.floor(x);<a name="line.126"></a>
130 <FONT color="green">127</FONT> if (fl == x) {<a name="line.127"></a>
131 <FONT color="green">128</FONT> return this.probability((int) x);<a name="line.128"></a>
132 <FONT color="green">129</FONT> } else {<a name="line.129"></a>
133 <FONT color="green">130</FONT> return 0;<a name="line.130"></a>
134 <FONT color="green">131</FONT> }<a name="line.131"></a>
135 <FONT color="green">132</FONT> }<a name="line.132"></a>
136 <FONT color="green">133</FONT> <a name="line.133"></a>
137 <FONT color="green">134</FONT> /**<a name="line.134"></a>
138 <FONT color="green">135</FONT> * For a random variable X whose values are distributed according<a name="line.135"></a>
139 <FONT color="green">136</FONT> * to this distribution, this method returns P(x0 &amp;le; X &amp;le; x1).<a name="line.136"></a>
140 <FONT color="green">137</FONT> *<a name="line.137"></a>
141 <FONT color="green">138</FONT> * @param x0 the inclusive, lower bound<a name="line.138"></a>
142 <FONT color="green">139</FONT> * @param x1 the inclusive, upper bound<a name="line.139"></a>
143 <FONT color="green">140</FONT> * @return the cumulative probability.<a name="line.140"></a>
144 <FONT color="green">141</FONT> * @throws MathException if the cumulative probability can not be<a name="line.141"></a>
145 <FONT color="green">142</FONT> * computed due to convergence or other numerical errors.<a name="line.142"></a>
146 <FONT color="green">143</FONT> * @throws IllegalArgumentException if x0 &gt; x1<a name="line.143"></a>
147 <FONT color="green">144</FONT> */<a name="line.144"></a>
148 <FONT color="green">145</FONT> public double cumulativeProbability(int x0, int x1) throws MathException {<a name="line.145"></a>
149 <FONT color="green">146</FONT> if (x0 &gt; x1) {<a name="line.146"></a>
150 <FONT color="green">147</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.147"></a>
151 <FONT color="green">148</FONT> WRONG_ORDER_ENDPOINTS_MESSAGE, x0, x1);<a name="line.148"></a>
152 <FONT color="green">149</FONT> }<a name="line.149"></a>
153 <FONT color="green">150</FONT> return cumulativeProbability(x1) - cumulativeProbability(x0 - 1);<a name="line.150"></a>
154 <FONT color="green">151</FONT> }<a name="line.151"></a>
155 <FONT color="green">152</FONT> <a name="line.152"></a>
156 <FONT color="green">153</FONT> /**<a name="line.153"></a>
157 <FONT color="green">154</FONT> * For a random variable X whose values are distributed according<a name="line.154"></a>
158 <FONT color="green">155</FONT> * to this distribution, this method returns the largest x, such<a name="line.155"></a>
159 <FONT color="green">156</FONT> * that P(X &amp;le; x) &amp;le; &lt;code&gt;p&lt;/code&gt;.<a name="line.156"></a>
160 <FONT color="green">157</FONT> *<a name="line.157"></a>
161 <FONT color="green">158</FONT> * @param p the desired probability<a name="line.158"></a>
162 <FONT color="green">159</FONT> * @return the largest x such that P(X &amp;le; x) &lt;= p<a name="line.159"></a>
163 <FONT color="green">160</FONT> * @throws MathException if the inverse cumulative probability can not be<a name="line.160"></a>
164 <FONT color="green">161</FONT> * computed due to convergence or other numerical errors.<a name="line.161"></a>
165 <FONT color="green">162</FONT> * @throws IllegalArgumentException if p &lt; 0 or p &gt; 1<a name="line.162"></a>
166 <FONT color="green">163</FONT> */<a name="line.163"></a>
167 <FONT color="green">164</FONT> public int inverseCumulativeProbability(final double p) throws MathException{<a name="line.164"></a>
168 <FONT color="green">165</FONT> if (p &lt; 0.0 || p &gt; 1.0) {<a name="line.165"></a>
169 <FONT color="green">166</FONT> throw MathRuntimeException.createIllegalArgumentException(<a name="line.166"></a>
170 <FONT color="green">167</FONT> OUT_OF_RANGE_POINT, p, 0.0, 1.0);<a name="line.167"></a>
171 <FONT color="green">168</FONT> }<a name="line.168"></a>
172 <FONT color="green">169</FONT> <a name="line.169"></a>
173 <FONT color="green">170</FONT> // by default, do simple bisection.<a name="line.170"></a>
174 <FONT color="green">171</FONT> // subclasses can override if there is a better method.<a name="line.171"></a>
175 <FONT color="green">172</FONT> int x0 = getDomainLowerBound(p);<a name="line.172"></a>
176 <FONT color="green">173</FONT> int x1 = getDomainUpperBound(p);<a name="line.173"></a>
177 <FONT color="green">174</FONT> double pm;<a name="line.174"></a>
178 <FONT color="green">175</FONT> while (x0 &lt; x1) {<a name="line.175"></a>
179 <FONT color="green">176</FONT> int xm = x0 + (x1 - x0) / 2;<a name="line.176"></a>
180 <FONT color="green">177</FONT> pm = checkedCumulativeProbability(xm);<a name="line.177"></a>
181 <FONT color="green">178</FONT> if (pm &gt; p) {<a name="line.178"></a>
182 <FONT color="green">179</FONT> // update x1<a name="line.179"></a>
183 <FONT color="green">180</FONT> if (xm == x1) {<a name="line.180"></a>
184 <FONT color="green">181</FONT> // this can happen with integer division<a name="line.181"></a>
185 <FONT color="green">182</FONT> // simply decrement x1<a name="line.182"></a>
186 <FONT color="green">183</FONT> --x1;<a name="line.183"></a>
187 <FONT color="green">184</FONT> } else {<a name="line.184"></a>
188 <FONT color="green">185</FONT> // update x1 normally<a name="line.185"></a>
189 <FONT color="green">186</FONT> x1 = xm;<a name="line.186"></a>
190 <FONT color="green">187</FONT> }<a name="line.187"></a>
191 <FONT color="green">188</FONT> } else {<a name="line.188"></a>
192 <FONT color="green">189</FONT> // update x0<a name="line.189"></a>
193 <FONT color="green">190</FONT> if (xm == x0) {<a name="line.190"></a>
194 <FONT color="green">191</FONT> // this can happen with integer division<a name="line.191"></a>
195 <FONT color="green">192</FONT> // simply increment x0<a name="line.192"></a>
196 <FONT color="green">193</FONT> ++x0;<a name="line.193"></a>
197 <FONT color="green">194</FONT> } else {<a name="line.194"></a>
198 <FONT color="green">195</FONT> // update x0 normally<a name="line.195"></a>
199 <FONT color="green">196</FONT> x0 = xm;<a name="line.196"></a>
200 <FONT color="green">197</FONT> }<a name="line.197"></a>
201 <FONT color="green">198</FONT> }<a name="line.198"></a>
202 <FONT color="green">199</FONT> }<a name="line.199"></a>
203 <FONT color="green">200</FONT> <a name="line.200"></a>
204 <FONT color="green">201</FONT> // insure x0 is the correct critical point<a name="line.201"></a>
205 <FONT color="green">202</FONT> pm = checkedCumulativeProbability(x0);<a name="line.202"></a>
206 <FONT color="green">203</FONT> while (pm &gt; p) {<a name="line.203"></a>
207 <FONT color="green">204</FONT> --x0;<a name="line.204"></a>
208 <FONT color="green">205</FONT> pm = checkedCumulativeProbability(x0);<a name="line.205"></a>
209 <FONT color="green">206</FONT> }<a name="line.206"></a>
210 <FONT color="green">207</FONT> <a name="line.207"></a>
211 <FONT color="green">208</FONT> return x0;<a name="line.208"></a>
212 <FONT color="green">209</FONT> }<a name="line.209"></a>
213 <FONT color="green">210</FONT> <a name="line.210"></a>
214 <FONT color="green">211</FONT> /**<a name="line.211"></a>
215 <FONT color="green">212</FONT> * Computes the cumulative probablity function and checks for NaN values returned.<a name="line.212"></a>
216 <FONT color="green">213</FONT> * Throws MathException if the value is NaN. Wraps and rethrows any MathException encountered<a name="line.213"></a>
217 <FONT color="green">214</FONT> * evaluating the cumulative probability function in a FunctionEvaluationException. Throws<a name="line.214"></a>
218 <FONT color="green">215</FONT> * FunctionEvaluationException of the cumulative probability function returns NaN.<a name="line.215"></a>
219 <FONT color="green">216</FONT> *<a name="line.216"></a>
220 <FONT color="green">217</FONT> * @param argument input value<a name="line.217"></a>
221 <FONT color="green">218</FONT> * @return cumulative probability<a name="line.218"></a>
222 <FONT color="green">219</FONT> * @throws FunctionEvaluationException if a MathException occurs computing the cumulative probability<a name="line.219"></a>
223 <FONT color="green">220</FONT> */<a name="line.220"></a>
224 <FONT color="green">221</FONT> private double checkedCumulativeProbability(int argument) throws FunctionEvaluationException {<a name="line.221"></a>
225 <FONT color="green">222</FONT> double result = Double.NaN;<a name="line.222"></a>
226 <FONT color="green">223</FONT> try {<a name="line.223"></a>
227 <FONT color="green">224</FONT> result = cumulativeProbability(argument);<a name="line.224"></a>
228 <FONT color="green">225</FONT> } catch (MathException ex) {<a name="line.225"></a>
229 <FONT color="green">226</FONT> throw new FunctionEvaluationException(ex, argument, ex.getPattern(), ex.getArguments());<a name="line.226"></a>
230 <FONT color="green">227</FONT> }<a name="line.227"></a>
231 <FONT color="green">228</FONT> if (Double.isNaN(result)) {<a name="line.228"></a>
232 <FONT color="green">229</FONT> throw new FunctionEvaluationException(argument,<a name="line.229"></a>
233 <FONT color="green">230</FONT> "Discrete cumulative probability function returned NaN for argument {0}", argument);<a name="line.230"></a>
234 <FONT color="green">231</FONT> }<a name="line.231"></a>
235 <FONT color="green">232</FONT> return result;<a name="line.232"></a>
236 <FONT color="green">233</FONT> }<a name="line.233"></a>
237 <FONT color="green">234</FONT> <a name="line.234"></a>
238 <FONT color="green">235</FONT> /**<a name="line.235"></a>
239 <FONT color="green">236</FONT> * Access the domain value lower bound, based on &lt;code&gt;p&lt;/code&gt;, used to<a name="line.236"></a>
240 <FONT color="green">237</FONT> * bracket a PDF root. This method is used by<a name="line.237"></a>
241 <FONT color="green">238</FONT> * {@link #inverseCumulativeProbability(double)} to find critical values.<a name="line.238"></a>
242 <FONT color="green">239</FONT> *<a name="line.239"></a>
243 <FONT color="green">240</FONT> * @param p the desired probability for the critical value<a name="line.240"></a>
244 <FONT color="green">241</FONT> * @return domain value lower bound, i.e.<a name="line.241"></a>
245 <FONT color="green">242</FONT> * P(X &amp;lt; &lt;i&gt;lower bound&lt;/i&gt;) &amp;lt; &lt;code&gt;p&lt;/code&gt;<a name="line.242"></a>
246 <FONT color="green">243</FONT> */<a name="line.243"></a>
247 <FONT color="green">244</FONT> protected abstract int getDomainLowerBound(double p);<a name="line.244"></a>
248 <FONT color="green">245</FONT> <a name="line.245"></a>
249 <FONT color="green">246</FONT> /**<a name="line.246"></a>
250 <FONT color="green">247</FONT> * Access the domain value upper bound, based on &lt;code&gt;p&lt;/code&gt;, used to<a name="line.247"></a>
251 <FONT color="green">248</FONT> * bracket a PDF root. This method is used by<a name="line.248"></a>
252 <FONT color="green">249</FONT> * {@link #inverseCumulativeProbability(double)} to find critical values.<a name="line.249"></a>
253 <FONT color="green">250</FONT> *<a name="line.250"></a>
254 <FONT color="green">251</FONT> * @param p the desired probability for the critical value<a name="line.251"></a>
255 <FONT color="green">252</FONT> * @return domain value upper bound, i.e.<a name="line.252"></a>
256 <FONT color="green">253</FONT> * P(X &amp;lt; &lt;i&gt;upper bound&lt;/i&gt;) &amp;gt; &lt;code&gt;p&lt;/code&gt;<a name="line.253"></a>
257 <FONT color="green">254</FONT> */<a name="line.254"></a>
258 <FONT color="green">255</FONT> protected abstract int getDomainUpperBound(double p);<a name="line.255"></a>
259 <FONT color="green">256</FONT> }<a name="line.256"></a>
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