Scilab Function
Last update : 16/12/2004

cdfnbn - cumulative distribution function negative binomial distribution

### Calling Sequence

  `[P,Q]=cdfnbn("PQ",S,Xn,Pr,Ompr)`
  `[S]=cdfnbn("S",Xn,Pr,Ompr,P,Q)`
  `[Xn]=cdfnbn("Xn",Pr,Ompr,P,Q,S)`
  `[Pr,Ompr]=cdfnbn("PrOmpr",P,Q,S,Xn)`

### Parameters

- `P,Q,S,Xn,Pr,Ompr`: six real vectors of the same size.
- `P,Q (Q=1-P)` : The cumulation from 0 to S of the negative binomial distribution. Input range: [0,1].
- `S`: The upper limit of cumulation of the binomial distribution. There are F or fewer failures before the XNth success. Input range: [0, +infinity). Search range: [0, 1E300]
- `Xn`: The number of successes. Input range: [0, +infinity). Search range: [0, 1E300]
- `Pr`: The probability of success in each binomial trial. Input range: [0,1]. Search range: [0,1].
- `Ompr`: 1-PR Input range: [0,1]. Search range: [0,1] PR + OMPR = 1.0

### Description

Calculates any one parameter of the negative binomial distribution given values for the others.

The cumulative negative binomial distribution returns the probability that there will be F or fewer failures before the XNth success in binomial trials each of which has probability of success PR.

The individual term of the negative binomial is the probability of S failures before XN successes and is Choose `( S, XN+S-1 ) * PR^(XN) * (1-PR)^S`

Formula 26.5.26 of Abramowitz and Stegun, Handbook of Mathematical Functions (1966) is used to reduce calculation of the cumulative distribution function to that of an incomplete beta.

Computation of other parameters involve a seach for a value that produces the desired value of P. The search relies on the monotinicity of P with the other parameter.

From DCDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters (February, 1994) Barry W. Brown, James Lovato and Kathy Russell. The University of Texas.