/// <summary>
/// Returns the probability of an event occuring in k out of n trials, given
/// that the probability of the event occuring in any one trial is p.
/// </summary>
/// <param name="p">Probability that the event occurs in any one trial.</param>
/// <param name="n">Number of trials.</param>
/// <param name="k">Number of times the event occurs in n trials.</param>
/// <returns></returns>
public static double BinomialProbabilityDensityFunction(double p, int n, int k)
{
//# of combinations can exceed max double for n > 1020
if (n > 1020)
{
double logB = MMath.LogCombin(n, k) + k * Math.Log(p) + (n - k) * Math.Log(1 - p);
return(Math.Exp(logB));
}
return(MMath.Combinations(n, k) * Math.Pow(p, k) * Math.Pow(1 - p, n - k));
}
/// <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>
/// The Irwin-Hall distribution results from the sum on n independent standard uniform variables
/// </summary>
/// <param name="x">The value at which to evaluate the distribution.</param>
/// <param name="n">The number of standard uniform variables.</param>
public static double IrwinHallProbabilityDensityFunction(double x, int n)
{
if (x < 0 || x > n)
{
return(0.0);
}
double d = 0;
for (int i = 0; i <= n; i++)
{
d += Math.Pow(-1, i) * MMath.Combinations(n, i) * Math.Pow(x - i, n - 1) * Math.Sign(x - i);
}
d *= 0.5;
d /= MMath.Factorial(n - 1);
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);
}
/// <summary>
/// Returns the CDF of Student's t distribution.
/// </summary>
/// <param name="x">Value at which the distribution is evaluated.</param>
/// <param name="k">Degrees of freedom.</param>
public static double StudentsTCumulativeDensityFunction(double x, double k)
{
//Tested against Excel for
//df = 1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 100, 200, 30, 400, 500 and
//x = -7.0, -6.9, -6.8, ..., 7.0
//maximum discrepancy was less than 0.04%
if (double.IsNaN(x))
{
return(double.NaN);
}
if (x == 0)
{
return(0.5);
}
if (x > 0)
{
double x2 = k / (x * x + k);
return(1.0 - 0.5 * MMath.RegularizedIncompleteBetaFunction(x2, 0.5 * k, 0.5));
}
return(1.0 - StudentsTCumulativeDensityFunction(-x, k));
}
/// <summary>
/// The PDF of the Poisson distribution.
/// The probability of n iid events occuring in an interval, if the expected number of event is lambda.
/// </summary>
/// <param name="lambda">Equals both the mean and variance of the distribution.</param>
/// <param name="n">The number of events.</param>
public static double PoissonProbabilityDensityFunction(double lambda, int n)
{
return(Math.Pow(lambda, n) * Math.Exp(-lambda) / MMath.Factorial(n));
}