rndKMgam#

Purpose#

Computes Gamma pseudo-random numbers.

Format#

{ x, newstate } = rndKMgam(r, c, alpha, state)#
Parameters:

  • r (scalar) – number of rows of resulting matrix.

  • c (scalar) – number of columns of resulting matrix.

  • alpha (matrix or vector or scalar) – Shape argument for gamma distribution. ExE conformable with the row and column dimensions of the return matrix, r and c.

  • state (scalar or 500x1 vector) –

    scalar case

    state = starting seed value only. If -1, GAUSS computes the starting seed based on the system clock.

    500x1 vector case

    state = the state vector returned from a previous call to one of the rndKM random number functions.

Returns:

  • x (RxC matrix) – Gamma distributed random numbers.

  • newstate (500x1 vector) – the updated state.

Remarks#

The properties of the pseudo-random numbers in x are:

\[\begin{split}E(x) = \alpha\\ Var(x) = \alpha\\ x > 0\\ \alpha > 0\end{split}\]

To generate gamma(alpha, theta) pseudo-random numbers where theta is a scale parameter, multiply the result of rndKMgam() by theta.

Thus

z =  theta * rndgam(1, 1, alpha);

has the properties

\[\begin{split}E(z) = \alpha * \theta\\ Var(z) = \alpha * \theta^2\\ z > 0\\ \alpha > 0\\ \theta > 0\end{split}\]

r and c will be truncated to integers if necessary.

Examples#

// Generate a 3x2 matrix of gamma random numbers
// with shape alpha = 5, using a fixed seed for repeatable output
{ x, newstate } = rndKMgam(3, 2, 5, 12345);
print x;

The output is a 3x2 matrix of gamma-distributed values. The sample mean is approximately 5, consistent with the theoretical mean of alpha:

5.6131502        2.6675772
4.6622884        3.3561170
4.9309951        6.8404543

Technical Notes#

rndKMgam() uses the recur-with-carry KISS+Monster algorithm described in the rndKMi() Technical Notes.

Source#

randkm.src

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