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comparison libs/commons-math-2.1/docs/userguide/distribution.html @ 10:5f2c5fb36e93
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date | Tue, 04 Jan 2011 10:00:53 +0100 |
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149 <div class="section"><h2><a name="a8_Probability_Distributions"></a>8 Probability Distributions</h2> | |
150 <div class="section"><h3><a name="a8.1_Overview"></a>8.1 Overview</h3> | |
151 <p> | |
152 The distributions package provide a framework for some commonly used | |
153 probability distributions. | |
154 </p> | |
155 </div> | |
156 <div class="section"><h3><a name="a8.2_Distribution_Framework"></a>8.2 Distribution Framework</h3> | |
157 <p> | |
158 The distribution framework provides the means to compute probability density | |
159 function (PDF) probabilities and cumulative distribution function (CDF) | |
160 probabilities for common probability distributions. Along with the direct | |
161 computation of PDF and CDF probabilities, the framework also allows for the | |
162 computation of inverse PDF and inverse CDF values. | |
163 </p> | |
164 <p> | |
165 Using a distribution object, PDF and CDF probabilities are easily computed | |
166 using the <code>cumulativeProbability</code> methods. For a distribution <code>X</code>, | |
167 and a domain value, <code>x</code>, <code>cumulativeProbability</code> computes | |
168 <code>P(X <= x)</code> (i.e. the lower tail probability of <code>X</code>). | |
169 </p> | |
170 <div class="source"><pre>TDistribution t = new TDistributionImpl(29); | |
171 double lowerTail = t.cumulativeProbability(-2.656); // P(T <= -2.656) | |
172 double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T >= 2.75)</pre> | |
173 </div> | |
174 <p> | |
175 The inverse PDF and CDF values are just as easily computed using the | |
176 <code>inverseCumulativeProbability</code> methods. For a distribution <code>X</code>, | |
177 and a probability, <code>p</code>, <code>inverseCumulativeProbability</code> | |
178 computes the domain value <code>x</code>, such that: | |
179 <ul><li><code>P(X <= x) = p</code>, for continuous distributions</li> | |
180 <li><code>P(X <= x) <= p</code>, for discrete distributions</li> | |
181 </ul> | |
182 | |
183 Notice the different cases for continuous and discrete distributions. This is the result | |
184 of PDFs not being invertible functions. As such, for discrete distributions, an exact | |
185 domain value can not be returned. Only the "best" domain value. For Commons-Math, the "best" | |
186 domain value is determined by the largest domain value whose cumulative probability is | |
187 less-than or equal to the given probability. | |
188 </p> | |
189 </div> | |
190 <div class="section"><h3><a name="a8.3_User_Defined_Distributions"></a>8.3 User Defined Distributions</h3> | |
191 <p> | |
192 Since there are numerous distributions and Commons-Math only directly supports a handful, | |
193 it may be necessary to extend the distribution framework to satisfy individual needs. It | |
194 is recommended that the <code>Distribution</code>, <code>ContinuousDistribution</code>, | |
195 <code>DiscreteDistribution</code>, and <code>IntegerDistribution</code> interfaces serve as | |
196 base types for any extension. These serve as the basis for all the distributions directly | |
197 supported by Commons-Math and using those interfaces for implementation purposes will | |
198 insure any extension is compatible with the remainder of Commons-Math. To aid in | |
199 implementing a distribution extension, the <code>AbstractDistribution</code>, | |
200 <code>AbstractContinuousDistribution</code>, and <code>AbstractIntegerDistribution</code> | |
201 provide implementation building blocks and offer a lot of default distribution | |
202 functionality. By extending these abstract classes directly, a good portion of the | |
203 repetitive distribution implementation is already developed and should save time and effort | |
204 in developing user defined distributions. | |
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