scipy.stats.burr12#
- scipy.stats.burr12 = <scipy.stats._continuous_distns.burr12_gen object>[source]#
- A Burr (Type XII) continuous random variable. - As an instance of the - rv_continuousclass,- burr12object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.- Notes - The probability density function for - burr12is:\[f(x; c, d) = c d \frac{x^{c-1}} {(1 + x^c)^{d + 1}}\]- for \(x >= 0\) and \(c, d > 0\). - burr12takes- cand- das shape parameters for \(c\) and \(d\).- This is the PDF corresponding to the twelfth CDF given in Burr’s list; specifically, it is equation (20) in Burr’s paper [1]. - The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- burr12.pdf(x, c, d, loc, scale)is identically equivalent to- burr12.pdf(y, c, d) / scalewith- y = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.- The Burr type 12 distribution is also sometimes referred to as the Singh-Maddala distribution from NIST [2]. - References [1]- Burr, I. W. “Cumulative frequency functions”, Annals of Mathematical Statistics, 13(2), pp 215-232 (1942). [3]- “Burr distribution”, https://en.wikipedia.org/wiki/Burr_distribution - Examples - >>> import numpy as np >>> from scipy.stats import burr12 >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Calculate the first four moments: - >>> c, d = 10, 4 >>> mean, var, skew, kurt = burr12.stats(c, d, moments='mvsk') - Display the probability density function ( - pdf):- >>> x = np.linspace(burr12.ppf(0.01, c, d), ... burr12.ppf(0.99, c, d), 100) >>> ax.plot(x, burr12.pdf(x, c, d), ... 'r-', lw=5, alpha=0.6, label='burr12 pdf') - Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed. - Freeze the distribution and display the frozen - pdf:- >>> rv = burr12(c, d) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') - Check accuracy of - cdfand- ppf:- >>> vals = burr12.ppf([0.001, 0.5, 0.999], c, d) >>> np.allclose([0.001, 0.5, 0.999], burr12.cdf(vals, c, d)) True - Generate random numbers: - >>> r = burr12.rvs(c, d, size=1000) - And compare the histogram: - >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> ax.legend(loc='best', frameon=False) >>> plt.show()   - Methods - rvs(c, d, loc=0, scale=1, size=1, random_state=None) - Random variates. - pdf(x, c, d, loc=0, scale=1) - Probability density function. - logpdf(x, c, d, loc=0, scale=1) - Log of the probability density function. - cdf(x, c, d, loc=0, scale=1) - Cumulative distribution function. - logcdf(x, c, d, loc=0, scale=1) - Log of the cumulative distribution function. - sf(x, c, d, loc=0, scale=1) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(x, c, d, loc=0, scale=1) - Log of the survival function. - ppf(q, c, d, loc=0, scale=1) - Percent point function (inverse of - cdf— percentiles).- isf(q, c, d, loc=0, scale=1) - Inverse survival function (inverse of - sf).- moment(order, c, d, loc=0, scale=1) - Non-central moment of the specified order. - stats(c, d, loc=0, scale=1, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(c, d, loc=0, scale=1) - (Differential) entropy of the RV. - fit(data) - Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments. - expect(func, args=(c, d), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) - Expected value of a function (of one argument) with respect to the distribution. - median(c, d, loc=0, scale=1) - Median of the distribution. - mean(c, d, loc=0, scale=1) - Mean of the distribution. - var(c, d, loc=0, scale=1) - Variance of the distribution. - std(c, d, loc=0, scale=1) - Standard deviation of the distribution. - interval(confidence, c, d, loc=0, scale=1) - Confidence interval with equal areas around the median.