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Probability (Colt 1.2.0 - API Specification)
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cern.jet.stat</FONT>
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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"><B>cern.jet.stat.Probability</B>
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<DL>
<DT>public class <B>Probability</B><DT>extends <A HREF="../../../cern/jet/math/Constants.html" title="class in cern.jet.math">Constants</A></DL>
<P>
Custom tailored numerical integration of certain probability distributions.
<p>
<b>Implementation:</b>
<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.
<P>
<P>
<DL>
<DT><B>Version:</B></DT>
<DD>0.91, 08-Dec-99</DD>
</DL>
<HR>
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<B>Method Summary</B></FONT></TD>
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<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#beta(double, double, double)">beta</A></B>(double a,
double b,
double x)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#betaComplemented(double, double, double)">betaComplemented</A></B>(double a,
double b,
double x)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#binomial(int, int, double)">binomial</A></B>(int k,
int n,
double p)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#binomialComplemented(int, int, double)">binomialComplemented</A></B>(int k,
int n,
double p)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#chiSquare(double, double)">chiSquare</A></B>(double v,
double x)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#chiSquareComplemented(double, double)">chiSquareComplemented</A></B>(double v,
double x)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#errorFunction(double)">errorFunction</A></B>(double x)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#errorFunctionComplemented(double)">errorFunctionComplemented</A></B>(double a)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#gamma(double, double, double)">gamma</A></B>(double a,
double b,
double x)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#gammaComplemented(double, double, double)">gammaComplemented</A></B>(double a,
double b,
double x)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#negativeBinomial(int, int, double)">negativeBinomial</A></B>(int k,
int n,
double p)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#negativeBinomialComplemented(int, int, double)">negativeBinomialComplemented</A></B>(int k,
int n,
double p)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#normal(double)">normal</A></B>(double a)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#normal(double, double, double)">normal</A></B>(double mean,
double variance,
double x)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#normalInverse(double)">normalInverse</A></B>(double y0)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#poisson(int, double)">poisson</A></B>(int k,
double mean)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#poissonComplemented(int, double)">poissonComplemented</A></B>(int k,
double mean)</CODE>
<BR>
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%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#studentT(double, double)">studentT</A></B>(double k,
double t)</CODE>
<BR>
Returns the integral from minus infinity to <tt>t</tt> of the Student-t
distribution with <tt>k > 0</tt> degrees of freedom.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE>static double</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../cern/jet/stat/Probability.html#studentTInverse(double, int)">studentTInverse</A></B>(double alpha,
int size)</CODE>
<BR>
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>
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<B>Method Detail</B></FONT></TD>
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<A NAME="beta(double, double, double)"><!-- --></A><H3>
beta</H3>
<PRE>
public static double <B>beta</B>(double a,
double b,
double 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>;
<P>
<DD><DL>
</DL>
</DD>
</DL>
<HR>
<A NAME="betaComplemented(double, double, double)"><!-- --></A><H3>
betaComplemented</H3>
<PRE>
public static double <B>betaComplemented</B>(double a,
double b,
double 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>.
<P>
<DD><DL>
</DL>
</DD>
</DL>
<HR>
<A NAME="binomial(int, int, double)"><!-- --></A><H3>
binomial</H3>
<PRE>
public static double <B>binomial</B>(int k,
int n,
double 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
<p>
<tt>y = binomial( k, n, p ) = Gamma.incompleteBeta( n-k, k+1, 1-p )</tt>.
<p>
All arguments must be positive,
<P>
<DD><DL>
<DT><B>Parameters:</B><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 <B>binomialComplemented</B>(int k,
int n,
double 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
<p>
<tt>y = binomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n-k, p )</tt>.
<p>
All arguments must be positive,
<P>
<DD><DL>
<DT><B>Parameters:</B><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 <B>chiSquare</B>(double v,
double 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.
<p>
The incomplete gamma integral is used, according to the
formula
<p>
<tt>y = chiSquare( v, x ) = incompleteGamma( v/2.0, x/2.0 )</tt>.
<p>
The arguments must both be positive.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>v</CODE> - degrees of freedom.<DD><CODE>x</CODE> - integration end point.
<DT><B>Throws:</B>
<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 <B>chiSquareComplemented</B>(double v,
double 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.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>v</CODE> - degrees of freedom.
<DT><B>Throws:</B>
<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 <B>errorFunction</B>(double 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>
<b>Implementation:</b>
For <tt>0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2)</tt>; otherwise
<tt>erf(x) = 1 - erfc(x)</tt>.
<p>
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).
<P>
<DD><DL>
<DT><B>Throws:</B>
<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 <B>errorFunctionComplemented</B>(double 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>
<b>Implementation:</b>
For small x, <tt>erfc(x) = 1 - erf(x)</tt>; otherwise rational
approximations are computed.
<p>
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).
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>a</CODE> - the argument to the function.
<DT><B>Throws:</B>
<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 <B>gamma</B>(double a,
double b,
double 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>.
<P>
<DD><DL>
<DT><B>Parameters:</B><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 <B>gammaComplemented</B>(double a,
double b,
double 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
<p>
y = Gamma.incompleteGammaComplement( b, a*x ).
<P>
<DD><DL>
<DT><B>Parameters:</B><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 <B>negativeBinomial</B>(int k,
int n,
double 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.
<p>
The terms are not computed individually; instead the incomplete
beta integral is employed, according to the formula
<p>
<tt>y = negativeBinomial( k, n, p ) = Gamma.incompleteBeta( n, k+1, p )</tt>.
All arguments must be positive,
<P>
<DD><DL>
<DT><B>Parameters:</B><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 <B>negativeBinomialComplemented</B>(int k,
int n,
double 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
<p>
y = negativeBinomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n, 1-p ).
All arguments must be positive,
<P>
<DD><DL>
<DT><B>Parameters:</B><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="normal(double)"><!-- --></A><H3>
normal</H3>
<PRE>
public static double <B>normal</B>(double 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>.
<P>
<DD><DL>
<DT><B>Throws:</B>
<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="normal(double, double, double)"><!-- --></A><H3>
normal</H3>
<PRE>
public static double <B>normal</B>(double mean,
double variance,
double 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>.
<P>
<DD><DL>
<DT><B>Parameters:</B><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><B>Throws:</B>
<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 <B>normalInverse</B>(double 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>.
<p>
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>.
<P>
<DD><DL>
<DT><B>Throws:</B>
<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="poisson(int, double)"><!-- --></A><H3>
poisson</H3>
<PRE>
public static double <B>poisson</B>(int k,
double 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
<p>
<tt>y = poisson( k, m ) = Gamma.incompleteGammaComplement( k+1, m )</tt>.
The arguments must both be positive.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>k</CODE> - number of terms.<DD><CODE>mean</CODE> - the mean of the poisson distribution.
<DT><B>Throws:</B>
<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 <B>poissonComplemented</B>(int k,
double 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
<p>
<tt>y = poissonComplemented( k, m ) = Gamma.incompleteGamma( k+1, m )</tt>.
The arguments must both be positive.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>k</CODE> - start term.<DD><CODE>mean</CODE> - the mean of the poisson distribution.
<DT><B>Throws:</B>
<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 <B>studentT</B>(double k,
double 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 > 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:
<p>
<tt>1 - studentT(k,t) = 0.5 * Gamma.incompleteBeta( k/2, 1/2, z )</tt>
where <tt>z = k/(k + t**2)</tt>.
<p>
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>.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>k</CODE> - degrees of freedom.<DD><CODE>t</CODE> - integration end point.
<DT><B>Throws:</B>
<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 <B>studentTInverse</B>(double alpha,
int 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>t<sub>alpha[size]</sub></tt>.
<p>
The function uses the studentT function to determine the return
value iteratively.
<P>
<DD><DL>
<DT><B>Parameters:</B><DD><CODE>alpha</CODE> - probability<DD><CODE>size</CODE> - size of data set</DL>
</DD>
</DL>
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