/// <inheritdoc />
/// <param name="k">The order of moment to compute.</param>
public override double RawMoment(int k)
{
if (k < 0)
{
throw new ArgumentOutOfRangeException(nameof(k));
}
else if (k == 0)
{
return(1.0);
}
else
{
// The (falling) factorial moments of the negative binomial distribution
// are F_k = (p/q)^m (r)_m, where (r)_m is a rising factorial.
// We can use this plus the Stirling numbers, which convert factorial
// to raw moments.
double[] s = AdvancedIntegerMath.StirlingNumbers2(k);
double f = p / q;
double t = 1.0;
double M = 0.0;
for (int i = 1; i <= k; i++)
{
t *= f * (r + i - 1);
M += s[i] * t;
}
return(M);
}
}
public void StirlingNumbers2RowSum()
{
foreach (int n in TestUtilities.GenerateIntegerValues(1, 100, 4))
{
double sum = 0.0;
foreach (double s in AdvancedIntegerMath.StirlingNumbers2(n))
{
sum += s;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(sum, AdvancedIntegerMath.BellNumber(n)));
}
}
public void StirlingNumberMatrixInverse()
{
int n = 8;
SquareMatrix S1 = new SquareMatrix(n);
for (int i = 0; i < n; i++)
{
double[] s = AdvancedIntegerMath.StirlingNumbers1(i);
for (int j = 0; j < s.Length; j++)
{
if ((i - j) % 2 == 0)
{
S1[i, j] = s[j];
}
else
{
S1[i, j] = -s[j];
}
}
}
SquareMatrix S2 = new SquareMatrix(n);
for (int i = 0; i < n; i++)
{
double[] s = AdvancedIntegerMath.StirlingNumbers2(i);
for (int j = 0; j < s.Length; j++)
{
S2[i, j] = s[j];
}
}
SquareMatrix S12 = S1 * S2;
SquareMatrix I = new SquareMatrix(n);
for (int i = 0; i < n; i++)
{
I[i, i] = 1.0;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(S12, I));
}
/// <inheritdoc />
public override double RawMoment(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException(nameof(r));
}
else if (r == 0)
{
return(1.0);
}
else
{
if (n < 2 * r)
{
// If n is small, just do a weighted sum over the support.
return(ExpectationValue(delegate(int k) { return (MoreMath.Pow(k, r)); }));
}
else
{
// If n >> r, convert analytically known factorial moments to raw moments
// using Sterling numbers of the 2nd kind. The (falling) factorial moment
// F_r = (n)_r p^r
// where (n)_r is a falling factorial.
double[] s = AdvancedIntegerMath.StirlingNumbers2(r);
double t = 1.0;
double M = 0.0;
for (int k = 1; k <= r; k++)
{
// s[0] = 0 (except for r=0, in which case we have already returned above)
t *= (n - (k - 1)) * p;
M += s[k] * t;
}
return(M);
}
}
}