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13 <html xmlns="http://www.w3.org/1999/xhtml">
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14 <head>
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15 <title>Math - The Commons Math User Guide - Statistics</title>
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24 <body class="composite">
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25 <div id="banner">
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26 <span id="bannerLeft">
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27
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28 Commons Math User Guide
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29
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30 </span>
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65
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66
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67 <h5>User Guide</h5>
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68 <ul>
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69
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70 <li class="none">
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71 <a href="../userguide/index.html">Contents</a>
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72 </li>
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73
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74 <li class="none">
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75 <a href="../userguide/overview.html">Overview</a>
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76 </li>
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77
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78 <li class="none">
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79 <a href="../userguide/stat.html">Statistics</a>
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80 </li>
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81
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82 <li class="none">
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83 <a href="../userguide/random.html">Data Generation</a>
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84 </li>
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85
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86 <li class="none">
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87 <a href="../userguide/linear.html">Linear Algebra</a>
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88 </li>
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89
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90 <li class="none">
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91 <a href="../userguide/analysis.html">Numerical Analysis</a>
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92 </li>
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93
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94 <li class="none">
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95 <a href="../userguide/special.html">Special Functions</a>
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96 </li>
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97
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99 <a href="../userguide/utilities.html">Utilities</a>
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100 </li>
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101
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102 <li class="none">
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103 <a href="../userguide/complex.html">Complex Numbers</a>
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104 </li>
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105
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106 <li class="none">
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107 <strong>Distributions</strong>
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108 </li>
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109
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111 <a href="../userguide/fraction.html">Fractions</a>
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115 <a href="../userguide/transform.html">Transform Methods</a>
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116 </li>
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117
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118 <li class="none">
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119 <a href="../userguide/geometry.html">3D Geometry</a>
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120 </li>
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121
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122 <li class="none">
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123 <a href="../userguide/optimization.html">Optimization</a>
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124 </li>
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125
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126 <li class="none">
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127 <a href="../userguide/ode.html">Ordinary Differential Equations</a>
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128 </li>
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129
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130 <li class="none">
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131 <a href="../userguide/genetics.html">Genetic Algorithms</a>
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132 </li>
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133 </ul>
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136 </a>
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145 </div>
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146 </div>
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147 <div id="bodyColumn">
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148 <div id="contentBox">
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149 <div class="section"><h2><a name="a8_Probability_Distributions"></a>8 Probability Distributions</h2>
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150 <div class="section"><h3><a name="a8.1_Overview"></a>8.1 Overview</h3>
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151 <p>
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152 The distributions package provide a framework for some commonly used
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153 probability distributions.
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154 </p>
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155 </div>
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156 <div class="section"><h3><a name="a8.2_Distribution_Framework"></a>8.2 Distribution Framework</h3>
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157 <p>
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158 The distribution framework provides the means to compute probability density
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159 function (PDF) probabilities and cumulative distribution function (CDF)
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160 probabilities for common probability distributions. Along with the direct
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161 computation of PDF and CDF probabilities, the framework also allows for the
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162 computation of inverse PDF and inverse CDF values.
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163 </p>
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164 <p>
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165 Using a distribution object, PDF and CDF probabilities are easily computed
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166 using the <code>cumulativeProbability</code> methods. For a distribution <code>X</code>,
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167 and a domain value, <code>x</code>, <code>cumulativeProbability</code> computes
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168 <code>P(X <= x)</code> (i.e. the lower tail probability of <code>X</code>).
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169 </p>
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170 <div class="source"><pre>TDistribution t = new TDistributionImpl(29);
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171 double lowerTail = t.cumulativeProbability(-2.656); // P(T <= -2.656)
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172 double upperTail = 1.0 - t.cumulativeProbability(2.75); // P(T >= 2.75)</pre>
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173 </div>
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174 <p>
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175 The inverse PDF and CDF values are just as easily computed using the
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176 <code>inverseCumulativeProbability</code> methods. For a distribution <code>X</code>,
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177 and a probability, <code>p</code>, <code>inverseCumulativeProbability</code>
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178 computes the domain value <code>x</code>, such that:
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179 <ul><li><code>P(X <= x) = p</code>, for continuous distributions</li>
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180 <li><code>P(X <= x) <= p</code>, for discrete distributions</li>
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181 </ul>
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182
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183 Notice the different cases for continuous and discrete distributions. This is the result
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184 of PDFs not being invertible functions. As such, for discrete distributions, an exact
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185 domain value can not be returned. Only the "best" domain value. For Commons-Math, the "best"
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186 domain value is determined by the largest domain value whose cumulative probability is
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187 less-than or equal to the given probability.
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188 </p>
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189 </div>
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190 <div class="section"><h3><a name="a8.3_User_Defined_Distributions"></a>8.3 User Defined Distributions</h3>
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191 <p>
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192 Since there are numerous distributions and Commons-Math only directly supports a handful,
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193 it may be necessary to extend the distribution framework to satisfy individual needs. It
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194 is recommended that the <code>Distribution</code>, <code>ContinuousDistribution</code>,
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195 <code>DiscreteDistribution</code>, and <code>IntegerDistribution</code> interfaces serve as
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196 base types for any extension. These serve as the basis for all the distributions directly
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197 supported by Commons-Math and using those interfaces for implementation purposes will
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198 insure any extension is compatible with the remainder of Commons-Math. To aid in
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199 implementing a distribution extension, the <code>AbstractDistribution</code>,
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200 <code>AbstractContinuousDistribution</code>, and <code>AbstractIntegerDistribution</code>
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201 provide implementation building blocks and offer a lot of default distribution
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202 functionality. By extending these abstract classes directly, a good portion of the
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203 repetitive distribution implementation is already developed and should save time and effort
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204 in developing user defined distributions.
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205 </p>
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206 </div>
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207 </div>
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209 </div>
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210 </div>
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211 <div class="clear">
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212 <hr/>
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213 </div>
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214 <div id="footer">
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215 <div class="xright">©
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216 2003-2010
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229 </div>
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230 </div>
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231 </body>
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232 </html>
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