scipy.special.nbdtr#
- scipy.special.nbdtr(k, n, p, out=None) = <ufunc 'nbdtr'>#
- Negative binomial cumulative distribution function. - Returns the sum of the terms 0 through k of the negative binomial distribution probability mass function, \[F = \sum_{j=0}^k {{n + j - 1}\choose{j}} p^n (1 - p)^j.\]- In a sequence of Bernoulli trials with individual success probabilities p, this is the probability that k or fewer failures precede the nth success. - Parameters:
- karray_like
- The maximum number of allowed failures (nonnegative int). 
- narray_like
- The target number of successes (positive int). 
- parray_like
- Probability of success in a single event (float). 
- outndarray, optional
- Optional output array for the function results 
 
- Returns:
- Fscalar or ndarray
- The probability of k or fewer failures before n successes in a sequence of events with individual success probability p. 
 
 - See also - nbdtrc
- Negative binomial survival function 
- nbdtrik
- Negative binomial quantile function 
- scipy.stats.nbinom
- Negative binomial distribution 
 - Notes - If floating point values are passed for k or n, they will be truncated to integers. - The terms are not summed directly; instead the regularized incomplete beta function is employed, according to the formula, \[\mathrm{nbdtr}(k, n, p) = I_{p}(n, k + 1).\]- Wrapper for the Cephes [1] routine - nbdtr.- The negative binomial distribution is also available as - scipy.stats.nbinom. Using- nbdtrdirectly can improve performance compared to the- cdfmethod of- scipy.stats.nbinom(see last example).- References [1]- Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ - Examples - Compute the function for - k=10and- n=5at- p=0.5.- >>> import numpy as np >>> from scipy.special import nbdtr >>> nbdtr(10, 5, 0.5) 0.940765380859375 - Compute the function for - n=10and- p=0.5at several points by providing a NumPy array or list for k.- >>> nbdtr([5, 10, 15], 10, 0.5) array([0.15087891, 0.58809853, 0.88523853]) - Plot the function for four different parameter sets. - >>> import matplotlib.pyplot as plt >>> k = np.arange(130) >>> n_parameters = [20, 20, 20, 80] >>> p_parameters = [0.2, 0.5, 0.8, 0.5] >>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot'] >>> parameters_list = list(zip(p_parameters, n_parameters, ... linestyles)) >>> fig, ax = plt.subplots(figsize=(8, 8)) >>> for parameter_set in parameters_list: ... p, n, style = parameter_set ... nbdtr_vals = nbdtr(k, n, p) ... ax.plot(k, nbdtr_vals, label=rf"$n={n},\, p={p}$", ... ls=style) >>> ax.legend() >>> ax.set_xlabel("$k$") >>> ax.set_title("Negative binomial cumulative distribution function") >>> plt.show()   - The negative binomial distribution is also available as - scipy.stats.nbinom. Using- nbdtrdirectly can be much faster than calling the- cdfmethod of- scipy.stats.nbinom, especially for small arrays or individual values. To get the same results one must use the following parametrization:- nbinom(n, p).cdf(k)=nbdtr(k, n, p).- >>> from scipy.stats import nbinom >>> k, n, p = 5, 3, 0.5 >>> nbdtr_res = nbdtr(k, n, p) # this will often be faster than below >>> stats_res = nbinom(n, p).cdf(k) >>> stats_res, nbdtr_res # test that results are equal (0.85546875, 0.85546875) - nbdtrcan evaluate different parameter sets by providing arrays with shapes compatible for broadcasting for k, n and p. Here we compute the function for three different k at four locations p, resulting in a 3x4 array.- >>> k = np.array([[5], [10], [15]]) >>> p = np.array([0.3, 0.5, 0.7, 0.9]) >>> k.shape, p.shape ((3, 1), (4,)) - >>> nbdtr(k, 5, p) array([[0.15026833, 0.62304687, 0.95265101, 0.9998531 ], [0.48450894, 0.94076538, 0.99932777, 0.99999999], [0.76249222, 0.99409103, 0.99999445, 1. ]])