scipy.special.nbdtrik

scipy.special.nbdtrik(y, n, p) = <ufunc 'nbdtrik'>

Inverse of nbdtr vs k.

Returns the inverse with respect to the parameter k of y = nbdtr(k, n, p), the negative binomial cumulative distribution function.

Parameters

yarray_like

The probability of k or fewer failures before n successes (float).

narray_like

The target number of successes (positive int).

parray_like

Probability of success in a single event (float).

Returns

kndarray

The maximum number of allowed failures such that nbdtr(k, n, p) = y.

See also

nbdtr

Cumulative distribution function of the negative binomial.

nbdtri

Inverse with respect to p of nbdtr(k, n, p).

nbdtrin

Inverse with respect to n of nbdtr(k, n, p).

Notes

Wrapper for the CDFLIB [1] Fortran routine cdfnbn.

Formula 26.5.26 of [2],

\[\sum_{j=k + 1}^\infty {{n + j - 1}\choose{j}} p^n (1 - p)^j = I_{1 - p}(k + 1, n),\]

is used to reduce calculation of the cumulative distribution function to that of a regularized incomplete beta \(I\).

Computation of k involves a search for a value that produces the desired value of y. The search relies on the monotonicity of y with k.

References

1(1,2)

Barry Brown, James Lovato, and Kathy Russell, CDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters.

2(1,2)

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.