/// <summary>
/// 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
/// <para>
/// y = negativeBinomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n, 1-p ).
///
/// All arguments must be positive,
/// </para>
/// </summary>
/// <param name="k"> end term. </param>
/// <param name="n"> the number of trials. </param>
/// <param name="p"> the probability of success (must be in <tt>(0.0,1.0)</tt>). </param>
/// <returns> result </returns>
public static double negativeBinomialComplemented(int k, int n, double p)
{
if ((p < 0.0) || (p > 1.0))
{
throw new System.ArgumentException();
}
if (k < 0)
{
return(0.0);
}
return(GammaFunctions.incompleteBeta(k + 1, n, 1.0 - p));
}
/// <summary>
/// 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:
/// <para>
/// <tt>1 - studentT(k,t) = 0.5 * Gamma.incompleteBeta( k/2, 1/2, z )</tt>
/// where <tt>z = k/(k + t**2)</tt>.
/// </para>
/// <para>
/// 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>.
///
/// </para>
/// </summary>
/// <param name="k"> degrees of freedom. </param>
/// <param name="t"> integration end point. </param>
/// <returns> result </returns>
/// <exception cref="ArithmeticException"> error </exception>
//JAVA TO C# CONVERTER WARNING: Method 'throws' clauses are not available in .NET:
//ORIGINAL LINE: public static double studentT(double k, double t) throws ArithmeticException
public static double studentT(double k, double t)
{
if (k <= 0)
{
throw new System.ArgumentException();
}
if (t == 0)
{
return(0.5);
}
double cdf = 0.5 * GammaFunctions.incompleteBeta(0.5 * k, 0.5, k / (k + t * t));
if (t >= 0)
{
cdf = 1.0 - cdf; // fixes bug reported by [email protected]
}
return(cdf);
}
/// <summary>
/// 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
/// <para>
/// <tt>y = binomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n-k, p )</tt>.
/// </para>
/// <para>
/// All arguments must be positive,
/// </para>
/// </summary>
/// <param name="k"> end term. </param>
/// <param name="n"> the number of trials. </param>
/// <param name="p"> the probability of success (must be in <tt>(0.0,1.0)</tt>). </param>
/// <returns> result </returns>
/// <exception cref="ArithmeticException"> error </exception>
public static double binomialComplemented(int k, int n, double p)
{
if ((p < 0.0) || (p > 1.0))
{
throw new System.ArgumentException();
}
if ((k < 0) || (n < k))
{
throw new System.ArgumentException();
}
if (k == n)
{
return(0.0);
}
if (k == 0)
{
return(1.0 - Math.Pow(1.0 - p, n - k));
}
return(GammaFunctions.incompleteBeta(k + 1, n - k, p));
}
/// <summary>
/// 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>;
/// </summary>
/// <param name="a"> a </param>
/// <param name="b"> b </param>
/// <param name="x"> x </param>
/// <returns> result </returns>
/// <exception cref="ArithmeticException"> error </exception>
public static double beta(double a, double b, double x)
{
return(GammaFunctions.incompleteBeta(a, b, x));
}
/// <summary>
/// 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>.
/// </summary>
/// <param name="a"> a </param>
/// <param name="b"> b </param>
/// <param name="x"> x </param>
/// <returns> result </returns>
/// <exception cref="ArithmeticException"> error </exception>
public static double betaComplemented(double a, double b, double x)
{
return(GammaFunctions.incompleteBeta(b, a, x));
}
