build/colt/doc/api/cern/jet/stat/Probability.html (a20104a) - Rabbit simulation · Portfolio Code

Probability.html

<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">

<HTML>
<HEAD>

<TITLE>
Probability (Colt 1.2.0 - API Specification)
</TITLE>

<SCRIPT type="text/javascript">
function windowTitle()
{
 parent.document.title="Probability (Colt 1.2.0 - API Specification)";
}
</SCRIPT>

</HEAD>

<A NAME="navbar_top"></A>
<A HREF="#skip-navbar_top" title="Skip navigation links"></A>
<TABLE BORDER="0" WIDTH="100%" CELLPADDING="1" CELLSPACING="0" SUMMARY="">
<TR>
<TD COLSPAN=3 BGCOLOR="#EEEEFF" CLASS="NavBarCell1">
<A NAME="navbar_top_firstrow"></A>
<TABLE BORDER="0" CELLPADDING="0" CELLSPACING="3" SUMMARY="">
 <TR ALIGN="center" VALIGN="top">
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../overview-summary.html">Overview</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-summary.html">Package</A>&nbsp;</TD>
 <TD BGCOLOR="#FFFFFF" CLASS="NavBarCell1Rev"> &nbsp;Class&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="class-use/Probability.html">Use</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-tree.html">Tree</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../deprecated-list.html">Deprecated</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../index-files/index-1.html">Index</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../help-doc.html">Help</A>&nbsp;</TD>
 </TR>
</TABLE>
</TD>
<TD ALIGN="right" VALIGN="top" ROWSPAN=3>
Colt 1.2.0
</TD>
</TR>

<TR>
<TD BGCOLOR="white" CLASS="NavBarCell2">
&nbsp;<A HREF="../../../cern/jet/stat/Gamma.html" title="class in cern.jet.stat">PREV CLASS</A>&nbsp;
&nbsp;NEXT CLASS</TD>
<TD BGCOLOR="white" CLASS="NavBarCell2">
 <A HREF="../../../index.html" target="_top">FRAMES</A> &nbsp;
&nbsp;<A HREF="Probability.html" target="_top">NO FRAMES</A> &nbsp;
&nbsp;<SCRIPT type="text/javascript">
 
</SCRIPT>
<NOSCRIPT>
 <A HREF="../../../allclasses-noframe.html">All Classes</A>
</NOSCRIPT>

</TD>
</TR>
<TR>
<TD VALIGN="top" CLASS="NavBarCell3">
 SUMMARY:&nbsp;NESTED&nbsp;|&nbsp;FIELD&nbsp;|&nbsp;CONSTR&nbsp;|&nbsp;<A HREF="#method_summary">METHOD</A></TD>
<TD VALIGN="top" CLASS="NavBarCell3">
DETAIL:&nbsp;FIELD&nbsp;|&nbsp;CONSTR&nbsp;|&nbsp;<A HREF="#method_detail">METHOD</A></TD>
</TR>
</TABLE>
<A NAME="skip-navbar_top"></A>

<HR>

<H2>

cern.jet.stat
 
Class Probability</H2>
<PRE>
<A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html" title="class or interface in java.lang">java.lang.Object</A>
 <IMG SRC="../../../resources/inherit.gif" ALT="extended by"><A HREF="../../../cern/jet/math/Constants.html" title="class in cern.jet.math">cern.jet.math.Constants</A>
 <IMG SRC="../../../resources/inherit.gif" ALT="extended by">cern.jet.stat.Probability
</PRE>
<HR>
<DL>
<DT>public class Probability<DT>extends <A HREF="../../../cern/jet/math/Constants.html" title="class in cern.jet.math">Constants</A></DL>

Custom tailored numerical integration of certain probability distributions.
 
 Implementation:
 <dt>
 Some code taken and adapted from the <A HREF="http://www.sci.usq.edu.au/staff/leighb/graph/Top.html">Java 2D Graph Package 2.4</A>,
 which in turn is a port from the <A HREF="http://people.ne.mediaone.net/moshier/index.html#Cephes">Cephes 2.2</A> Math Library (C).
 Most Cephes code (missing from the 2D Graph Package) directly ported.

<DL>
<DT>Version:</DT>
 <DD>0.91, 08-Dec-99</DD>
</DL>
<HR>

<A NAME="method_summary"></A>
<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor">
<TD COLSPAN=2>
Method Summary</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#beta(double, double, double)">beta</A>(double&nbsp;a,
 double&nbsp;b,
 double&nbsp;x)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the area from zero to <tt>x</tt> under the beta density
 function.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#betaComplemented(double, double, double)">betaComplemented</A>(double&nbsp;a,
 double&nbsp;b,
 double&nbsp;x)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the area under the right hand tail (from <tt>x</tt> to
 infinity) of the beta density function.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#binomial(int, int, double)">binomial</A>(int&nbsp;k,
 int&nbsp;n,
 double&nbsp;p)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the sum of the terms <tt>0</tt> through <tt>k</tt> of the Binomial
 probability density.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#binomialComplemented(int, int, double)">binomialComplemented</A>(int&nbsp;k,
 int&nbsp;n,
 double&nbsp;p)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the sum of the terms <tt>k+1</tt> through <tt>n</tt> of the Binomial
 probability density.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#chiSquare(double, double)">chiSquare</A>(double&nbsp;v,
 double&nbsp;x)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the area under the left hand tail (from 0 to <tt>x</tt>)
 of the Chi square probability density function with
 <tt>v</tt> degrees of freedom.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#chiSquareComplemented(double, double)">chiSquareComplemented</A>(double&nbsp;v,
 double&nbsp;x)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the area under the right hand tail (from <tt>x</tt> to
 infinity) of the Chi square probability density function
 with <tt>v</tt> degrees of freedom.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#errorFunction(double)">errorFunction</A>(double&nbsp;x)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the error function of the normal distribution; formerly named <tt>erf</tt>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#errorFunctionComplemented(double)">errorFunctionComplemented</A>(double&nbsp;a)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the complementary Error function of the normal distribution; formerly named <tt>erfc</tt>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#gamma(double, double, double)">gamma</A>(double&nbsp;a,
 double&nbsp;b,
 double&nbsp;x)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the integral from zero to <tt>x</tt> of the gamma probability
 density function.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#gammaComplemented(double, double, double)">gammaComplemented</A>(double&nbsp;a,
 double&nbsp;b,
 double&nbsp;x)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the integral from <tt>x</tt> to infinity of the gamma
 probability density function:
 </TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#negativeBinomial(int, int, double)">negativeBinomial</A>(int&nbsp;k,
 int&nbsp;n,
 double&nbsp;p)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the sum of the terms <tt>0</tt> through <tt>k</tt> of the Negative Binomial Distribution.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#negativeBinomialComplemented(int, int, double)">negativeBinomialComplemented</A>(int&nbsp;k,
 int&nbsp;n,
 double&nbsp;p)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the sum of the terms <tt>k+1</tt> to infinity of the Negative
 Binomial distribution.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#normal(double)">normal</A>(double&nbsp;a)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the area under the Normal (Gaussian) probability density
 function, integrated from minus infinity to <tt>x</tt> (assumes mean is zero, variance is one).</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#normal(double, double, double)">normal</A>(double&nbsp;mean,
 double&nbsp;variance,
 double&nbsp;x)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the area under the Normal (Gaussian) probability density
 function, integrated from minus infinity to <tt>x</tt>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#normalInverse(double)">normalInverse</A>(double&nbsp;y0)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the value, <tt>x</tt>, for which the area under the
 Normal (Gaussian) probability density function (integrated from
 minus infinity to <tt>x</tt>) is equal to the argument <tt>y</tt> (assumes mean is zero, variance is one); formerly named <tt>ndtri</tt>.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#poisson(int, double)">poisson</A>(int&nbsp;k,
 double&nbsp;mean)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the sum of the first <tt>k</tt> terms of the Poisson distribution.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#poissonComplemented(int, double)">poissonComplemented</A>(int&nbsp;k,
 double&nbsp;mean)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the sum of the terms <tt>k+1</tt> to <tt>Infinity</tt> of the Poisson distribution.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#studentT(double, double)">studentT</A>(double&nbsp;k,
 double&nbsp;t)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the integral from minus infinity to <tt>t</tt> of the Student-t 
 distribution with <tt>k &gt; 0</tt> degrees of freedom.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%">
<CODE>static&nbsp;double</CODE></TD>
<TD><CODE><A HREF="../../../cern/jet/stat/Probability.html#studentTInverse(double, int)">studentTInverse</A>(double&nbsp;alpha,
 int&nbsp;size)</CODE>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Returns the value, <tt>t</tt>, for which the area under the
 Student-t probability density function (integrated from
 minus infinity to <tt>t</tt>) is equal to <tt>1-alpha/2</tt>.</TD>
</TR>
</TABLE>
&nbsp;<A NAME="methods_inherited_from_class_java.lang.Object"></A>
<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#EEEEFF" CLASS="TableSubHeadingColor">
<TD>Methods inherited from class java.lang.<A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html" title="class or interface in java.lang">Object</A></TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#equals(java.lang.Object)" title="class or interface in java.lang">equals</A>, <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#getClass()" title="class or interface in java.lang">getClass</A>, <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#hashCode()" title="class or interface in java.lang">hashCode</A>, <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#notify()" title="class or interface in java.lang">notify</A>, <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#notifyAll()" title="class or interface in java.lang">notifyAll</A>, <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#toString()" title="class or interface in java.lang">toString</A>, <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#wait()" title="class or interface in java.lang">wait</A>, <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#wait(long)" title="class or interface in java.lang">wait</A>, <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/Object.html#wait(long, int)" title="class or interface in java.lang">wait</A></CODE></TD>
</TR>
</TABLE>
&nbsp;

<A NAME="method_detail"></A>
<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#CCCCFF" CLASS="TableHeadingColor">
<TD COLSPAN=1>
Method Detail</TD>
</TR>
</TABLE>

<A NAME="beta(double, double, double)"></A><H3>
beta</H3>
<PRE>
public static double beta(double&nbsp;a,
 double&nbsp;b,
 double&nbsp;x)</PRE>
<DL>
<DD>Returns the area from zero to <tt>x</tt> under the beta density
 function.
 <pre>
 x
 - -
 | (a+b) | | a-1 b-1
 P(x) = ---------- | t (1-t) dt
 - - | |
 | (a) | (b) -
 0
 </pre>
 This function is identical to the incomplete beta
 integral function <tt>Gamma.incompleteBeta(a, b, x)</tt>.

The complemented function is

<tt>1 - P(1-x) = Gamma.incompleteBeta( b, a, x )</tt>;

<DD><DL>
</DL>
</DD>
</DL>
<HR>

<A NAME="betaComplemented(double, double, double)"></A><H3>
betaComplemented</H3>
<PRE>
public static double betaComplemented(double&nbsp;a,
 double&nbsp;b,
 double&nbsp;x)</PRE>
<DL>
<DD>Returns the area under the right hand tail (from <tt>x</tt> to
 infinity) of the beta density function.
 
 This function is identical to the incomplete beta
 integral function <tt>Gamma.incompleteBeta(b, a, x)</tt>.

<DD><DL>
</DL>
</DD>
</DL>
<HR>

<A NAME="binomial(int, int, double)"></A><H3>
binomial</H3>
<PRE>
public static double binomial(int&nbsp;k,
 int&nbsp;n,
 double&nbsp;p)</PRE>
<DL>
<DD>Returns the sum of the terms <tt>0</tt> through <tt>k</tt> of the Binomial
 probability density.
 <pre>
 k
 -- ( n ) j n-j
 > ( ) p (1-p)
 -- ( j )
 j=0
 </pre>
 The terms are not summed directly; instead the incomplete
 beta integral is employed, according to the formula
 
 <tt>y = binomial( k, n, p ) = Gamma.incompleteBeta( n-k, k+1, 1-p )</tt>.
 
 All arguments must be positive,

<DD><DL>
<DT>Parameters:<DD><CODE>k</CODE> - end term.<DD><CODE>n</CODE> - the number of trials.<DD><CODE>p</CODE> - the probability of success (must be in <tt>(0.0,1.0)</tt>).</DL>
</DD>
</DL>
<HR>

<A NAME="binomialComplemented(int, int, double)"></A><H3>
binomialComplemented</H3>
<PRE>
public static double binomialComplemented(int&nbsp;k,
 int&nbsp;n,
 double&nbsp;p)</PRE>
<DL>
<DD>Returns the sum of the terms <tt>k+1</tt> through <tt>n</tt> of the Binomial
 probability density.
 <pre>
 n
 -- ( n ) j n-j
 > ( ) p (1-p)
 -- ( j )
 j=k+1
 </pre>
 The terms are not summed directly; instead the incomplete
 beta integral is employed, according to the formula
 
 <tt>y = binomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n-k, p )</tt>.
 
 All arguments must be positive,

<DD><DL>
<DT>Parameters:<DD><CODE>k</CODE> - end term.<DD><CODE>n</CODE> - the number of trials.<DD><CODE>p</CODE> - the probability of success (must be in <tt>(0.0,1.0)</tt>).</DL>
</DD>
</DL>
<HR>

<A NAME="chiSquare(double, double)"></A><H3>
chiSquare</H3>
<PRE>
public static double chiSquare(double&nbsp;v,
 double&nbsp;x)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the area under the left hand tail (from 0 to <tt>x</tt>)
 of the Chi square probability density function with
 <tt>v</tt> degrees of freedom.
 <pre>
 inf.
 -
 1 | | v/2-1 -t/2
 P( x | v ) = ----------- | t e dt
 v/2 - | |
 2 | (v/2) -
 x
 </pre>
 where <tt>x</tt> is the Chi-square variable.
 
 The incomplete gamma integral is used, according to the
 formula
 
 <tt>y = chiSquare( v, x ) = incompleteGamma( v/2.0, x/2.0 )</tt>.
 
 The arguments must both be positive.

<DD><DL>
<DT>Parameters:<DD><CODE>v</CODE> - degrees of freedom.<DD><CODE>x</CODE> - integration end point.
<DT>Throws:
<DD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></CODE></DL>
</DD>
</DL>
<HR>

<A NAME="chiSquareComplemented(double, double)"></A><H3>
chiSquareComplemented</H3>
<PRE>
public static double chiSquareComplemented(double&nbsp;v,
 double&nbsp;x)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the area under the right hand tail (from <tt>x</tt> to
 infinity) of the Chi square probability density function
 with <tt>v</tt> degrees of freedom.
 <pre>
 inf.
 -
 1 | | v/2-1 -t/2
 P( x | v ) = ----------- | t e dt
 v/2 - | |
 2 | (v/2) -
 x
 </pre>
 where <tt>x</tt> is the Chi-square variable.

The incomplete gamma integral is used, according to the
 formula

<tt>y = chiSquareComplemented( v, x ) = incompleteGammaComplement( v/2.0, x/2.0 )</tt>.

The arguments must both be positive.

<DD><DL>
<DT>Parameters:<DD><CODE>v</CODE> - degrees of freedom.
<DT>Throws:
<DD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></CODE></DL>
</DD>
</DL>
<HR>

<A NAME="errorFunction(double)"></A><H3>
errorFunction</H3>
<PRE>
public static double errorFunction(double&nbsp;x)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the error function of the normal distribution; formerly named <tt>erf</tt>.
 The integral is
 <pre>
 x 
 -
 2 | | 2
 erf(x) = -------- | exp( - t ) dt.
 sqrt(pi) | |
 -
 0
 </pre>
 Implementation:
 For <tt>0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2)</tt>; otherwise
 <tt>erf(x) = 1 - erfc(x)</tt>.
 
 Code adapted from the <A HREF="http://www.sci.usq.edu.au/staff/leighb/graph/Top.html">Java 2D Graph Package 2.4</A>,
 which in turn is a port from the <A HREF="http://people.ne.mediaone.net/moshier/index.html#Cephes">Cephes 2.2</A> Math Library (C).

<DD><DL>

<DT>Throws:
<DD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></CODE></DL>
</DD>
</DL>
<HR>

<A NAME="errorFunctionComplemented(double)"></A><H3>
errorFunctionComplemented</H3>
<PRE>
public static double errorFunctionComplemented(double&nbsp;a)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the complementary Error function of the normal distribution; formerly named <tt>erfc</tt>.
 <pre>
 1 - erf(x) =

inf. 
 -
 2 | | 2
 erfc(x) = -------- | exp( - t ) dt
 sqrt(pi) | |
 -
 x
 </pre>
 Implementation:
 For small x, <tt>erfc(x) = 1 - erf(x)</tt>; otherwise rational
 approximations are computed.
 
 Code adapted from the <A HREF="http://www.sci.usq.edu.au/staff/leighb/graph/Top.html">Java 2D Graph Package 2.4</A>,
 which in turn is a port from the <A HREF="http://people.ne.mediaone.net/moshier/index.html#Cephes">Cephes 2.2</A> Math Library (C).

<DD><DL>
<DT>Parameters:<DD><CODE>a</CODE> - the argument to the function.
<DT>Throws:
<DD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></CODE></DL>
</DD>
</DL>
<HR>

<A NAME="gamma(double, double, double)"></A><H3>
gamma</H3>
<PRE>
public static double gamma(double&nbsp;a,
 double&nbsp;b,
 double&nbsp;x)</PRE>
<DL>
<DD>Returns the integral from zero to <tt>x</tt> of the gamma probability
 density function.
 <pre>
 x
 b -
 a | | b-1 -at
 y = ----- | t e dt
 - | |
 | (b) -
 0
 </pre>
 The incomplete gamma integral is used, according to the
 relation

<tt>y = Gamma.incompleteGamma( b, a*x )</tt>.

<DD><DL>
<DT>Parameters:<DD><CODE>a</CODE> - the paramater a (alpha) of the gamma distribution.<DD><CODE>b</CODE> - the paramater b (beta, lambda) of the gamma distribution.<DD><CODE>x</CODE> - integration end point.</DL>
</DD>
</DL>
<HR>

<A NAME="gammaComplemented(double, double, double)"></A><H3>
gammaComplemented</H3>
<PRE>
public static double gammaComplemented(double&nbsp;a,
 double&nbsp;b,
 double&nbsp;x)</PRE>
<DL>
<DD>Returns the integral from <tt>x</tt> to infinity of the gamma
 probability density function:
 <pre>
 inf.
 b -
 a | | b-1 -at
 y = ----- | t e dt
 - | |
 | (b) -
 x
 </pre>
 The incomplete gamma integral is used, according to the
 relation
 
 y = Gamma.incompleteGammaComplement( b, a*x ).

<DD><DL>
<DT>Parameters:<DD><CODE>a</CODE> - the paramater a (alpha) of the gamma distribution.<DD><CODE>b</CODE> - the paramater b (beta, lambda) of the gamma distribution.<DD><CODE>x</CODE> - integration end point.</DL>
</DD>
</DL>
<HR>

<A NAME="negativeBinomial(int, int, double)"></A><H3>
negativeBinomial</H3>
<PRE>
public static double negativeBinomial(int&nbsp;k,
 int&nbsp;n,
 double&nbsp;p)</PRE>
<DL>
<DD>Returns the sum of the terms <tt>0</tt> through <tt>k</tt> of the Negative Binomial Distribution.
 <pre>
 k
 -- ( n+j-1 ) n j
 > ( ) p (1-p)
 -- ( j )
 j=0
 </pre>
 In a sequence of Bernoulli trials, this is the probability
 that <tt>k</tt> or fewer failures precede the <tt>n</tt>-th success.
 
 The terms are not computed individually; instead the incomplete
 beta integral is employed, according to the formula
 
 <tt>y = negativeBinomial( k, n, p ) = Gamma.incompleteBeta( n, k+1, p )</tt>.

All arguments must be positive,

<DD><DL>
<DT>Parameters:<DD><CODE>k</CODE> - end term.<DD><CODE>n</CODE> - the number of trials.<DD><CODE>p</CODE> - the probability of success (must be in <tt>(0.0,1.0)</tt>).</DL>
</DD>
</DL>
<HR>

<A NAME="negativeBinomialComplemented(int, int, double)"></A><H3>
negativeBinomialComplemented</H3>
<PRE>
public static double negativeBinomialComplemented(int&nbsp;k,
 int&nbsp;n,
 double&nbsp;p)</PRE>
<DL>
<DD>Returns the sum of the terms <tt>k+1</tt> to infinity of the Negative
 Binomial distribution.
 <pre>
 inf
 -- ( n+j-1 ) n j
 > ( ) p (1-p)
 -- ( j )
 j=k+1
 </pre>
 The terms are not computed individually; instead the incomplete
 beta integral is employed, according to the formula
 
 y = negativeBinomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n, 1-p ).

<A NAME="normal(double)"></A><H3>
normal</H3>
<PRE>
public static double normal(double&nbsp;a)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the area under the Normal (Gaussian) probability density
 function, integrated from minus infinity to <tt>x</tt> (assumes mean is zero, variance is one).
 <pre>
 x
 -
 1 | | 2
 normal(x) = --------- | exp( - t /2 ) dt
 sqrt(2pi) | |
 -
 -inf.

= ( 1 + erf(z) ) / 2
 = erfc(z) / 2
 </pre>
 where <tt>z = x/sqrt(2)</tt>.
 Computation is via the functions <tt>errorFunction</tt> and <tt>errorFunctionComplement</tt>.

<DD><DL>

<A NAME="normal(double, double, double)"></A><H3>
normal</H3>
<PRE>
public static double normal(double&nbsp;mean,
 double&nbsp;variance,
 double&nbsp;x)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the area under the Normal (Gaussian) probability density
 function, integrated from minus infinity to <tt>x</tt>.
 <pre>
 x
 -
 1 | | 2
 normal(x) = --------- | exp( - (t-mean) / 2v ) dt
 sqrt(2pi*v)| |
 -
 -inf.

</pre>
 where <tt>v = variance</tt>.
 Computation is via the functions <tt>errorFunction</tt>.

<DD><DL>
<DT>Parameters:<DD><CODE>mean</CODE> - the mean of the normal distribution.<DD><CODE>variance</CODE> - the variance of the normal distribution.<DD><CODE>x</CODE> - the integration limit.
<DT>Throws:
<DD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></CODE></DL>
</DD>
</DL>
<HR>

<A NAME="normalInverse(double)"></A><H3>
normalInverse</H3>
<PRE>
public static double normalInverse(double&nbsp;y0)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the value, <tt>x</tt>, for which the area under the
 Normal (Gaussian) probability density function (integrated from
 minus infinity to <tt>x</tt>) is equal to the argument <tt>y</tt> (assumes mean is zero, variance is one); formerly named <tt>ndtri</tt>.
 
 For small arguments <tt>0 < y < exp(-2)</tt>, the program computes
 <tt>z = sqrt( -2.0 * log(y) )</tt>; then the approximation is
 <tt>x = z - log(z)/z - (1/z) P(1/z) / Q(1/z)</tt>.
 There are two rational functions P/Q, one for <tt>0 < y < exp(-32)</tt>
 and the other for <tt>y</tt> up to <tt>exp(-2)</tt>. 
 For larger arguments,
 <tt>w = y - 0.5</tt>, and <tt>x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2))</tt>.

<DD><DL>

<A NAME="poisson(int, double)"></A><H3>
poisson</H3>
<PRE>
public static double poisson(int&nbsp;k,
 double&nbsp;mean)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the sum of the first <tt>k</tt> terms of the Poisson distribution.
 <pre>
 k j
 -- -m m
 > e --
 -- j!
 j=0
 </pre>
 The terms are not summed directly; instead the incomplete
 gamma integral is employed, according to the relation
 
 <tt>y = poisson( k, m ) = Gamma.incompleteGammaComplement( k+1, m )</tt>.

The arguments must both be positive.

<DD><DL>
<DT>Parameters:<DD><CODE>k</CODE> - number of terms.<DD><CODE>mean</CODE> - the mean of the poisson distribution.
<DT>Throws:
<DD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></CODE></DL>
</DD>
</DL>
<HR>

<A NAME="poissonComplemented(int, double)"></A><H3>
poissonComplemented</H3>
<PRE>
public static double poissonComplemented(int&nbsp;k,
 double&nbsp;mean)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the sum of the terms <tt>k+1</tt> to <tt>Infinity</tt> of the Poisson distribution.
 <pre>
 inf. j
 -- -m m
 > e --
 -- j!
 j=k+1
 </pre>
 The terms are not summed directly; instead the incomplete
 gamma integral is employed, according to the formula
 
 <tt>y = poissonComplemented( k, m ) = Gamma.incompleteGamma( k+1, m )</tt>.

The arguments must both be positive.

<DD><DL>
<DT>Parameters:<DD><CODE>k</CODE> - start term.<DD><CODE>mean</CODE> - the mean of the poisson distribution.
<DT>Throws:
<DD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></CODE></DL>
</DD>
</DL>
<HR>

<A NAME="studentT(double, double)"></A><H3>
studentT</H3>
<PRE>
public static double studentT(double&nbsp;k,
 double&nbsp;t)
 throws <A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></PRE>
<DL>
<DD>Returns the integral from minus infinity to <tt>t</tt> of the Student-t 
 distribution with <tt>k &gt; 0</tt> degrees of freedom.
 <pre>
 t
 -
 | |
 - | 2 -(k+1)/2
 | ( (k+1)/2 ) | ( x )
 ---------------------- | ( 1 + --- ) dx
 - | ( k )
 sqrt( k pi ) | ( k/2 ) |
 | |
 -
 -inf.
 </pre>
 Relation to incomplete beta integral:
 
 <tt>1 - studentT(k,t) = 0.5 * Gamma.incompleteBeta( k/2, 1/2, z )</tt>
 where <tt>z = k/(k + t**2)</tt>.
 
 Since the function is symmetric about <tt>t=0</tt>, the area under the
 right tail of the density is found by calling the function
 with <tt>-t</tt> instead of <tt>t</tt>.

<DD><DL>
<DT>Parameters:<DD><CODE>k</CODE> - degrees of freedom.<DD><CODE>t</CODE> - integration end point.
<DT>Throws:
<DD><CODE><A HREF="http://java.sun.com/j2se/1.4/docs/api/java/lang/ArithmeticException.html" title="class or interface in java.lang">ArithmeticException</A></CODE></DL>
</DD>
</DL>
<HR>

<A NAME="studentTInverse(double, int)"></A><H3>
studentTInverse</H3>
<PRE>
public static double studentTInverse(double&nbsp;alpha,
 int&nbsp;size)</PRE>
<DL>
<DD>Returns the value, <tt>t</tt>, for which the area under the
 Student-t probability density function (integrated from
 minus infinity to <tt>t</tt>) is equal to <tt>1-alpha/2</tt>.
 The value returned corresponds to usual Student t-distribution lookup
 table for <tt>talpha[size]</tt>.
 
 The function uses the studentT function to determine the return
 value iteratively.

<DD><DL>
<DT>Parameters:<DD><CODE>alpha</CODE> - probability<DD><CODE>size</CODE> - size of data set</DL>
</DD>
</DL>

<HR>

<A NAME="navbar_bottom"></A>
<A HREF="#skip-navbar_bottom" title="Skip navigation links"></A>
<TABLE BORDER="0" WIDTH="100%" CELLPADDING="1" CELLSPACING="0" SUMMARY="">
<TR>
<TD COLSPAN=3 BGCOLOR="#EEEEFF" CLASS="NavBarCell1">
<A NAME="navbar_bottom_firstrow"></A>
<TABLE BORDER="0" CELLPADDING="0" CELLSPACING="3" SUMMARY="">
 <TR ALIGN="center" VALIGN="top">
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../overview-summary.html">Overview</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-summary.html">Package</A>&nbsp;</TD>
 <TD BGCOLOR="#FFFFFF" CLASS="NavBarCell1Rev"> &nbsp;Class&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="class-use/Probability.html">Use</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="package-tree.html">Tree</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../deprecated-list.html">Deprecated</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../index-files/index-1.html">Index</A>&nbsp;</TD>
 <TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../../../help-doc.html">Help</A>&nbsp;</TD>
 </TR>
</TABLE>
</TD>
<TD ALIGN="right" VALIGN="top" ROWSPAN=3>
Colt 1.2.0
</TD>
</TR>

</TD>
</TR>
<TR>
<TD VALIGN="top" CLASS="NavBarCell3">
 SUMMARY:&nbsp;NESTED&nbsp;|&nbsp;FIELD&nbsp;|&nbsp;CONSTR&nbsp;|&nbsp;<A HREF="#method_summary">METHOD</A></TD>
<TD VALIGN="top" CLASS="NavBarCell3">
DETAIL:&nbsp;FIELD&nbsp;|&nbsp;CONSTR&nbsp;|&nbsp;<A HREF="#method_detail">METHOD</A></TD>
</TR>
</TABLE>
<A NAME="skip-navbar_bottom"></A>

<HR>
Jump to the <a target=_top href=http://dsd.lbl.gov/~hoschek/colt >Colt Homepage</a>
</BODY>
</HTML>

Rabbit simulation

initial commit