/// <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>
/// For n independent draws from a standard uniform distribution (range = 0-1), the probability that the minimum is less than or equal to x.
/// </summary>
public static double IndependentStandardUniformMinimumCumulativeDistributionFunction(double x, int nDistributions)
{
double p = 0;
int sign = 1;
for (int i = 1; i < nDistributions; i++)
{
p += MMath.Combinations(nDistributions, i) * Math.Pow(x, i) * sign;
sign *= -1;
}
return p;
}
/// <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);
}