# scipy.stats.rel_breitwigner#

scipy.stats.rel_breitwigner = <scipy.stats._continuous_distns.rel_breitwigner_gen object>[source]#

A relativistic Breit-Wigner random variable.

As an instance of the rv_continuous class, rel_breitwigner object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

cauchy

Cauchy distribution, also known as the Breit-Wigner distribution.

Notes

The probability density function for rel_breitwigner is

$f(x, \rho) = \frac{k}{(x^2 - \rho^2)^2 + \rho^2}$

where

$k = \frac{2\sqrt{2}\rho^2\sqrt{\rho^2 + 1}} {\pi\sqrt{\rho^2 + \rho\sqrt{\rho^2 + 1}}}$

The relativistic Breit-Wigner distribution is used in high energy physics to model resonances . It gives the uncertainty in the invariant mass, $$M$$ , of a resonance with characteristic mass $$M_0$$ and decay-width $$\Gamma$$, where $$M$$, $$M_0$$ and $$\Gamma$$ are expressed in natural units. In SciPy’s parametrization, the shape parameter $$\rho$$ is equal to $$M_0/\Gamma$$ and takes values in $$(0, \infty)$$.

Equivalently, the relativistic Breit-Wigner distribution is said to give the uncertainty in the center-of-mass energy $$E_{\text{cm}}$$. In natural units, the speed of light $$c$$ is equal to 1 and the invariant mass $$M$$ is equal to the rest energy $$Mc^2$$. In the center-of-mass frame, the rest energy is equal to the total energy .

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, rel_breitwigner.pdf(x, rho, loc, scale) is identically equivalent to rel_breitwigner.pdf(y, rho) / scale with 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.

$$\rho = M/\Gamma$$ and $$\Gamma$$ is the scale parameter. For example, if one seeks to model the $$Z^0$$ boson with $$M_0 \approx 91.1876 \text{ GeV}$$ and $$\Gamma \approx 2.4952\text{ GeV}$$  one can set rho=91.1876/2.4952 and scale=2.4952.

To ensure a physically meaningful result when using the fit method, one should set floc=0 to fix the location parameter to 0.

References



Relativistic Breit-Wigner distribution, Wikipedia, https://en.wikipedia.org/wiki/Relativistic_Breit-Wigner_distribution



Invariant mass, Wikipedia, https://en.wikipedia.org/wiki/Invariant_mass



Center-of-momentum frame, Wikipedia, https://en.wikipedia.org/wiki/Center-of-momentum_frame



M. Tanabashi et al. (Particle Data Group) Phys. Rev. D 98, 030001 - Published 17 August 2018

Examples

>>> import numpy as np
>>> from scipy.stats import rel_breitwigner
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)


Calculate the first four moments:

>>> rho = 36.5
>>> mean, var, skew, kurt = rel_breitwigner.stats(rho, moments='mvsk')


Display the probability density function (pdf):

>>> x = np.linspace(rel_breitwigner.ppf(0.01, rho),
...                 rel_breitwigner.ppf(0.99, rho), 100)
>>> ax.plot(x, rel_breitwigner.pdf(x, rho),
...        'r-', lw=5, alpha=0.6, label='rel_breitwigner 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 = rel_breitwigner(rho)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')


Check accuracy of cdf and ppf:

>>> vals = rel_breitwigner.ppf([0.001, 0.5, 0.999], rho)
>>> np.allclose([0.001, 0.5, 0.999], rel_breitwigner.cdf(vals, rho))
True


Generate random numbers:

>>> r = rel_breitwigner.rvs(rho, size=1000)


And compare the histogram:

>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x, x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()


Methods

 rvs(rho, loc=0, scale=1, size=1, random_state=None) Random variates. pdf(x, rho, loc=0, scale=1) Probability density function. logpdf(x, rho, loc=0, scale=1) Log of the probability density function. cdf(x, rho, loc=0, scale=1) Cumulative distribution function. logcdf(x, rho, loc=0, scale=1) Log of the cumulative distribution function. sf(x, rho, loc=0, scale=1) Survival function (also defined as 1 - cdf, but sf is sometimes more accurate). logsf(x, rho, loc=0, scale=1) Log of the survival function. ppf(q, rho, loc=0, scale=1) Percent point function (inverse of cdf — percentiles). isf(q, rho, loc=0, scale=1) Inverse survival function (inverse of sf). moment(order, rho, loc=0, scale=1) Non-central moment of the specified order. stats(rho, loc=0, scale=1, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). entropy(rho, 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=(rho,), 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(rho, loc=0, scale=1) Median of the distribution. mean(rho, loc=0, scale=1) Mean of the distribution. var(rho, loc=0, scale=1) Variance of the distribution. std(rho, loc=0, scale=1) Standard deviation of the distribution. interval(confidence, rho, loc=0, scale=1) Confidence interval with equal areas around the median.