cdfGam

Purpose

Computes the regularized lower incomplete gamma function.

Format

p = cdfGam(x, intlim)

Parameters:

• x (NxK matrix) – Values at which to evaluate the regularized lower incomplete gamma function. \(x > 0\).

• int_lim (LxM matrix) – ExE compatible with x, containing the integration limit. \(int\_lim > 0\).

Returns:

p (matrix, max(N,L) by max(K,M)) – Each element in p is the regularized lower incomplete gamma function evaluated at the corresponding element in x.

Examples

Basic example

p = cdfGam(1.2, 3);

After the above code, P will equal

0.9287

Matrix example

// Create a 1x4 row vector
x = { 0.5 1 3 10 };

// Create a 6x1 column vector: 0, 0.2, 0.4, ..., 1.0
int_lim = seqa(0,.2,6);

/*
** Compute for all combinations of the elements
** of 'x' and 'int_lim'
*/
p = cdfGam(x, int_lim);

print "intlim = " int_lim;
print "p = " p;

After the code above:

intlim =
0.00000000
0.20000000
0.40000000
0.60000000
0.80000000
1.0000000

p =
0.00000000       0.00000000       0.00000000       0.00000000
0.47291074       0.18126925     0.0011484812   2.3530688e-014
0.62890663       0.32967995     0.0079263319   2.0098099e-011
0.72667832       0.45118836      0.023115288   9.6697183e-010
0.79409679       0.55067104      0.047422596   1.4331002e-008
0.84270079       0.63212056      0.080301397   1.1142548e-007

This computes the integrals over the range from 0 to 1, in increments of 0.2, at the parameter values 0.5, 1, 3, 10.

Remarks

The regularized lower incomplete gamma function returns the integral

\[\text{cdfGam(x, int_lim)} = \int_{0}^{int\_lim} \frac{e^{-t}t^{(x-1)}}{\Gamma(x)}dt\]

A -1 is returned for those elements with invalid inputs.