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# scipy.stats.ncf

scipy.stats.ncf = <scipy.stats._continuous_distns.ncf_gen object>

  A non-central F distribution continuous random variable.

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

  See also

  `scipy.stats.f`

    Fisher distribution

  Notes

  The probability density function for `ncf` is:

  \[\begin{split}f(x, n_1, n_2, \lambda) = \exp\left(\frac{\lambda}{2} + \lambda n_1 \frac{x}{2(n_1 x + n_2)} \right) n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\ (n_2 + n_1 x)^{-(n_1 + n_2)/2} \gamma(n_1/2) \gamma(1 + n_2/2) \\ \frac{L^{\frac{n_1}{2}-1}_{n_2/2} \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)} {B(n_1/2, n_2/2) \gamma\left(\frac{n_1 + n_2}{2}\right)}\end{split}\]

  for \(n_1, n_2 > 0\), \(\lambda \ge 0\). Here \(n_1\) is the degrees of freedom in the numerator, \(n_2\) the degrees of freedom in the denominator, \(\lambda\) the non-centrality parameter, \(\gamma\) is the logarithm of the Gamma function, \(L_n^k\) is a generalized Laguerre polynomial and \(B\) is the beta function.

  `ncf` takes `dfn`, `dfd` and `nc` as shape parameters. If `nc=0`, the distribution becomes equivalent to the Fisher distribution.

  This distribution uses routines from the Boost Math C++ library for the computation of the `pdf`, `cdf`, `ppf`, `stats`, `sf` and `isf` methods. [1]

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `ncf.pdf(x, dfn, dfd, nc, loc, scale)` is identically equivalent to `ncf.pdf(y, dfn, dfd, nc) / 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]

  The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

  Examples

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

  Get the support:

      >>> dfn, dfd, nc = 27, 27, 0.416
      >>> lb, ub = ncf.support(dfn, dfd, nc)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = ncf.stats(dfn, dfd, nc, moments='mvsk')

  Display the probability density function (`pdf`):

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

  Check accuracy of `cdf` and `ppf`:

      >>> vals = ncf.ppf([0.001, 0.5, 0.999], dfn, dfd, nc)
      >>> np.allclose([0.001, 0.5, 0.999], ncf.cdf(vals, dfn, dfd, nc))
      True

  Generate random numbers:

      >>> r = ncf.rvs(dfn, dfd, nc, 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()