# rndNegBinomial¶

## Purpose¶

Computes negative binomial pseudo-random numbers with a choice of underlying random number generator.

## Format¶

x = rndNegBinomial(r, c, ns, prob)
{ x, newstate } = rndNegBinomial(r, c, ns, prob, state)
Parameters:
• r (scalar) – number of rows of resulting matrix.

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

• ns (matrix, vector or scalar) – “event” argument for negative binomial distribution, scalar or ExE conformable matrix with r and c.

• prob (matrix, vector or scalar) – “probability” argument for negative binomial distribution, scalar or ExE conformable matrix with r and c.

• state (scalar or opaque vector) –

Optional argument.

scalar case

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

opaque vector case

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

Returns:
• x (RxC matrix) – negative binomial distributed random numbers.

• newstate (Opaque vector) – the updated state.

## Examples¶

### Example 1¶

Simulate the number of failures before 30 successes where each trial has a 70% probability of success.

num_obs = 100;

num_s = 30;
prob = 0.70;

num_f = rndNegBinomial(num_obs, 1, num_s, prob);


### Example 2¶

An alternative parameterization specifies the negative binomial distribution in terms of a dispersion parameter (dp) and a mean parameter (mu). If you would prefer to think of it in those terms, you may do so by passing in the dispersion parameter dp, in place of num_s and passing in $$dp/(dp + mu)$$ in place of prob.

// dispersion parameter
dp = 12;

// mean parameter
mu = 3;

x = rndNegBinomial(100, 1, dp, dp./(dp + mu));


## Remarks¶

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

\begin{align}\begin{aligned}\begin{split}E(x) = num\_s*\frac{1 - prob}{prob}\\\end{split}\\\begin{split}Var(x) = num\_s*\frac{1 - prob}{prob^2}\\\end{split}\\\begin{split}num\_s > 0\\\end{split}\\0 < prob < 1\end{aligned}\end{align}

rndNegBinomial() has a different parameterization than the deprecated rndnb(). To convert a call to rndnb() to an equivalent call to rndNegBinomial(), pass in $$1 - prob$$ in place of prob. For example, the following two calls are equivalent.

x_1 = rndnb(1e6, 1, 15, 0.3);
x_2 = rndNegBinomial(1e6, 1, 15, 0.7);


r and c will be truncated to integers if necessary.

## Technical Notes¶

The default generator for rndNegBinomial() is the SFMT Mersenne-Twister 19937. You can specify a different underlying random number generator with the function rndCreateState().