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def irwinhall(*args, **kwds)
An Irwin-Hall (Uniform Sum) continuous random variable.
An `Irwin-Hall <https://en.wikipedia.org/wiki/Irwin-Hall_distribution/>`_
continuous random variable is the sum of :math:`n` independent
standard uniform random variables [1]_ [2]_.
As an instance of the `rv_continuous` class, `irwinhall` 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(n, loc=0, scale=1, size=1, random_state=None)
Random variates.
pdf(x, n, loc=0, scale=1)
Probability density function.
logpdf(x, n, loc=0, scale=1)
Log of the probability density function.
cdf(x, n, loc=0, scale=1)
Cumulative distribution function.
logcdf(x, n, loc=0, scale=1)
Log of the cumulative distribution function.
sf(x, n, loc=0, scale=1)
Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(x, n, loc=0, scale=1)
Log of the survival function.
ppf(q, n, loc=0, scale=1)
Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, n, loc=0, scale=1)
Inverse survival function (inverse of ``sf``).
moment(order, n, loc=0, scale=1)
Non-central moment of the specified order.
stats(n, loc=0, scale=1, moments='mv')
Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(n, 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=(n,), 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(n, loc=0, scale=1)
Median of the distribution.
mean(n, loc=0, scale=1)
Mean of the distribution.
var(n, loc=0, scale=1)
Variance of the distribution.
std(n, loc=0, scale=1)
Standard deviation of the distribution.
interval(confidence, n, loc=0, scale=1)
Confidence interval with equal areas around the median.
Notes
-----
Applications include `Rao's Spacing Test
<https://jammalam.faculty.pstat.ucsb.edu/html/favorite/test.htm>`_,
a more powerful alternative to the Rayleigh test
when the data are not unimodal, and radar [3]_.
Conveniently, the pdf and cdf are the :math:`n`-fold convolution of
the ones for the standard uniform distribution, which is also the
definition of the cardinal B-splines of degree :math:`n-1`
having knots evenly spaced from :math:`1` to :math:`n` [4]_ [5]_.
The Bates distribution, which represents the *mean* of statistically
independent, uniformly distributed random variables, is simply the
Irwin-Hall distribution scaled by :math:`1/n`. For example, the frozen
distribution ``bates = irwinhall(10, scale=1/10)`` represents the
distribution of the mean of 10 uniformly distributed random variables.
The probability density above is defined in the "standardized" form. To shift
and/or scale the distribution use the ``loc`` and ``scale`` parameters.
Specifically, ``irwinhall.pdf(x, n, loc, scale)`` is identically
equivalent to ``irwinhall.pdf(y, n) / 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.
References
----------
.. [1] P. Hall, "The distribution of means for samples of size N drawn
from a population in which the variate takes values between 0 and 1,
all such values being equally probable",
Biometrika, Volume 19, Issue 3-4, December 1927, Pages 240-244,
:doi:`10.1093/biomet/19.3-4.240`.
.. [2] J. O. Irwin, "On the frequency distribution of the means of samples
from a population having any law of frequency with finite moments,
with special reference to Pearson's Type II,
Biometrika, Volume 19, Issue 3-4, December 1927, Pages 225-239,
:doi:`0.1093/biomet/19.3-4.225`.
.. [3] K. Buchanan, T. Adeyemi, C. Flores-Molina, S. Wheeland and D. Overturf,
"Sidelobe behavior and bandwidth characteristics
of distributed antenna arrays,"
2018 United States National Committee of
URSI National Radio Science Meeting (USNC-URSI NRSM),
Boulder, CO, USA, 2018, pp. 1-2.
https://www.usnc-ursi-archive.org/nrsm/2018/papers/B15-9.pdf.
.. [4] Amos Ron, "Lecture 1: Cardinal B-splines and convolution operators", p. 1
https://pages.cs.wisc.edu/~deboor/887/lec1new.pdf.
.. [5] Trefethen, N. (2012, July). B-splines and convolution. Chebfun.
Retrieved April 30, 2024, from http://www.chebfun.org/examples/approx/BSplineConv.html.
Examples
--------
>>> import numpy as np
>>> from scipy.stats import irwinhall
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> n = 10
>>> mean, var, skew, kurt = irwinhall.stats(n, moments='mvsk')
Display the probability density function (``pdf``):
>>> x = np.linspace(irwinhall.ppf(0.01, n),
... irwinhall.ppf(0.99, n), 100)
>>> ax.plot(x, irwinhall.pdf(x, n),
... 'r-', lw=5, alpha=0.6, label='irwinhall 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 = irwinhall(n)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of ``cdf`` and ``ppf``:
>>> vals = irwinhall.ppf([0.001, 0.5, 0.999], n)
>>> np.allclose([0.001, 0.5, 0.999], irwinhall.cdf(vals, n))
True
Generate random numbers:
>>> r = irwinhall.rvs(n, 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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