SciPy

numpy.random.RandomState.noncentral_chisquare

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

Draw samples from a noncentral chi-square distribution.

The noncentral \chi^2 distribution is a generalisation of the \chi^2 distribution.

Parameters :

df : int

Degrees of freedom, should be >= 1.

nonc : float

Non-centrality, should be > 0.

size : int or tuple of ints

Shape of the output.

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.

In Delhi (2007), it is noted that the noncentral chi-square is useful in bombing and coverage problems, the probability of killing the point target given by the noncentral chi-squared distribution.

References

[R155]Delhi, M.S. Holla, “On a noncentral chi-square distribution in the analysis of weapon systems effectiveness”, Metrika, Volume 15, Number 1 / December, 1970.
[R156]Wikipedia, “Noncentral chi-square distribution” http://en.wikipedia.org/wiki/Noncentral_chi-square_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, normed=True)
>>> plt.show()

(Source code, png, pdf)

../../_images/numpy-random-RandomState-noncentral_chisquare-1_00_00.png

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

(png, pdf)

../../_images/numpy-random-RandomState-noncentral_chisquare-1_01_00.png

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, normed=True)
>>> plt.show()

(png, pdf)

../../_images/numpy-random-RandomState-noncentral_chisquare-1_02_00.png