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# scipy.stats.irwinhall

scipy.stats.irwinhall = <scipy.stats._continuous_distns.irwinhall_gen object>

  An Irwin-Hall (Uniform Sum) continuous random variable.

  An Irwin-Hall continuous random variable is the sum of \(n\) 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 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, a more powerful alternative to the Rayleigh test when the data are not unimodal, and radar [3].

  Conveniently, the pdf and cdf are the \(n\)-fold convolution of the ones for the standard uniform distribution, which is also the definition of the cardinal B-splines of degree \(n-1\) having knots evenly spaced from \(1\) to \(n\) [4] [5].

  The Bates distribution, which represents the mean of statistically independent, uniformly distributed random variables, is simply the Irwin-Hall distribution scaled by \(1/n\). 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)

  Get the support:

      >>> n = 10
      >>> lb, ub = irwinhall.support(n)

  Calculate the first four moments:

      >>> 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()