/// <summary>
/// Returns the PDF of a beta distribution with shape parameters a and b.
/// f(x) = x^(a-1) * (1-x)^(b-1) / B(a,b)
/// </summary>
/// <param name="x">Value of the random variable for which the PDF is beign evaluated. x is between 0 and 1.</param>
/// <param name="a">Number of trials.</param>
/// <param name="b">Number of times the event occurs in n trials.</param>
/// <returns></returns>
public static double BetaProbabilityDensityFunction(double x, double a, double b)
{
if (x < 0 || x > 1)
{
throw new ArgumentException("x must be between 0 and 1");
}
double B = MMath.BetaFunction(a, b);
double d = Math.Pow(x, a - 1) * Math.Pow(1 - x, b - 1) / B;
return(d);
}
/// <summary>
/// Returns the PDF of the F distribution.
/// </summary>
/// <param name="x">Value at which the distribution is evaluated.</param>
/// <param name="k1">Degrees of freedom for numerator chi-sqared distribution. k1 > 0.</param>
/// <param name="k2">Degrees of freedom for denominator chi-sqared distribution. k2 > 0.</param>
public static double FProbabilityDensityFunction(double x, int k1, int k2)
{
if (k1 <= 0 || k2 <= 0)
{
throw new ArgumentException("k1 and k2 must be greater than 0.");
}
if (x == 0)
{
return(0.0);
}
double a1 = Math.Pow(k1 * x, 0.5 * k1);
double a2 = Math.Pow(k2, 0.5 * k2);
double a3 = Math.Pow(k1 * x + k2, 0.5 * (k1 + k2));
double a = a1 * a2 / a3;
double b = MMath.BetaFunction(0.5 * k1, 0.5 * k2);
double c = x * b;
return(a / c);
}