One Hat Cyber Team
Your IP :
216.73.216.115
Server IP :
194.44.31.54
Server :
Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software :
Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version :
5.6.40
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
usr
/
share
/
doc
/
Macaulay2
/
Probability
/
html
/
View File Name :
_density.html
<!DOCTYPE html> <html lang="en"> <head> <title>density -- probability density (or mass) function</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="basic probability functions" href="index.html">Probability</a> » <a title="probability distribution class" href="___Probability__Distribution.html">ProbabilityDistribution</a> » <a title="probability density function" href="___Density__Function.html">DensityFunction</a> » <a title="probability density (or mass) function" href="_density.html">density</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> next | previous | <a href="___Distribution__Function.html">forward</a> | <a href="___Density__Function.html">backward</a> | <a href="___Density__Function.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>density -- probability density (or mass) function</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">density_X x</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">X</span>, <span>a <a title="probability distribution class" href="___Probability__Distribution.html">probability distribution</a></span>, </span></li> <li><span><span class="tt">x</span>, <span>a <a title="the class of all real numbers" href="../../Macaulay2Doc/html/___R__R.html">real number</a></span>, </span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all real numbers" href="../../Macaulay2Doc/html/___R__R.html">real number</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>For a discrete probability distribution, this returns values of of the <a href="https://en.wikipedia.org/wiki/probability_mass_function">probability mass function</a> of the distribution, i.e., \(f_X(x) = P(X = x)\).</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : X = binomialDistribution(5, 0.25) o1 = B(5,.25) o1 : DiscreteProbabilityDistribution</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : density_X 2 o2 = .263671875 o2 : RR (of precision 53)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : binomial(5, 2) * 0.25^2 * 0.75^3 o3 = .263671875 o3 : RR (of precision 53)</code></pre> </td> </tr> </table> <div> <p>For a continuous probability distribution, this returns values of the <a href="https://en.wikipedia.org/wiki/probability_density_function">probability density function</a> of the distribution, i.e., the integrand in \(\int_a^b f_X(x)\,dx = P(a\leq X \leq b)\).</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : Z = normalDistribution() o4 = N(0,1) o4 : ContinuousProbabilityDistribution</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : density_Z 0 o5 = .3989422804014327 o5 : RR (of precision 53)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : 1/sqrt(2 * pi) o6 = .3989422804014327 o6 : RR (of precision 53)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : integrate(density_Z, -1, 1) o7 = .682689492137086 o7 : RR (of precision 53)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : integrate(density_Z, -2, 2) o8 = .9544997361034697 o8 : RR (of precision 53)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : integrate(density_Z, -3, 3) o9 = .997300204130616 o9 : RR (of precision 53)</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">density</span>:</h2> <ul> <li><kbd>density(DiscreteProbabilityDistribution,Number)</kbd></li> <li><kbd>density(ProbabilityDistribution,Number)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="probability density (or mass) function" href="_density.html">density</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Probability.m2:645:0</span>.</p> </div> </div> </div> </body> </html>