# numpy.random.noncentral_chisquare

random.noncentral_chisquare(df, nonc, size=None)

  Draw samples from a noncentral chi-square distribution.

  The noncentral \(\chi^2\) distribution is a generalization of the \(\chi^2\) distribution.

  Note

  New code should use the `noncentral_chisquare` method of a `default_rng()` instance instead; please see the Quick Start.

  Parameters
    df float or array_like of floats

      Degrees of freedom, must be > 0.

      Changed in version 1.10.0: Earlier NumPy versions required dfnum > 1.

    nonc float or array_like of floats

      Non-centrality, must be non-negative.

    size int or tuple of ints, optional

      Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. If size is `None` (default), a single value is returned if `df` and `nonc` are both scalars. Otherwise, `np.broadcast(df, nonc).size` samples are drawn.

  Returns
    out ndarray or scalar

      Drawn samples from the parameterized noncentral chi-square distribution.

  See also

  `random.Generator.noncentral_chisquare`

    which should be used for new code.

  Notes

  The probability density function for the noncentral Chi-square distribution is

  \[P(x;df,nonc) = \sum^{\infty}_{i=0} \frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x),\]

  where \(Y_{q}\) is the Chi-square with q degrees of freedom.

  References

  1

    Wikipedia, “Noncentral chi-squared distribution” https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

  Examples

  Draw values from the distribution and plot the histogram

      >>> import matplotlib.pyplot as plt
      >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
      ...                   bins=200, density=True)
      >>> plt.show()

  Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.

      >>> plt.figure()
      >>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
      ...                   bins=np.arange(0., 25, .1), density=True)
      >>> values2 = plt.hist(np.random.chisquare(3, 100000),
      ...                    bins=np.arange(0., 25, .1), density=True)
      >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
      >>> plt.show()

  Demonstrate how large values of non-centrality lead to a more symmetric distribution.

      >>> plt.figure()
      >>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
      ...                   bins=200, density=True)
      >>> plt.show()

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