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def rel_breitwigner(*args, **kwds)
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.
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 <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#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.
See Also
--------
cauchy: Cauchy distribution, also known as the Breit-Wigner distribution.
Notes
-----
The probability density function for `rel_breitwigner` is
.. math::
f(x, \rho) = \frac{k}{(x^2 - \rho^2)^2 + \rho^2}
where
.. math::
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 [1]_. It gives the uncertainty in the invariant mass,
:math:`M` [2]_, of a resonance with characteristic mass :math:`M_0` and
decay-width :math:`\Gamma`, where :math:`M`, :math:`M_0` and :math:`\Gamma`
are expressed in natural units. In SciPy's parametrization, the shape
parameter :math:`\rho` is equal to :math:`M_0/\Gamma` and takes values in
:math:`(0, \infty)`.
Equivalently, the relativistic Breit-Wigner distribution is said to give
the uncertainty in the center-of-mass energy :math:`E_{\text{cm}}`. In
natural units, the speed of light :math:`c` is equal to 1 and the invariant
mass :math:`M` is equal to the rest energy :math:`Mc^2`. In the
center-of-mass frame, the rest energy is equal to the total energy [3]_.
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.
:math:`\rho = M/\Gamma` and :math:`\Gamma` is the scale parameter. For
example, if one seeks to model the :math:`Z^0` boson with :math:`M_0
\approx 91.1876 \text{ GeV}` and :math:`\Gamma \approx 2.4952\text{ GeV}`
[4]_ 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
----------
.. [1] Relativistic Breit-Wigner distribution, Wikipedia,
https://en.wikipedia.org/wiki/Relativistic_Breit-Wigner_distribution
.. [2] Invariant mass, Wikipedia,
https://en.wikipedia.org/wiki/Invariant_mass
.. [3] Center-of-momentum frame, Wikipedia,
https://en.wikipedia.org/wiki/Center-of-momentum_frame
.. [4] 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[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
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