numpy.random.noncentral_chisquare — NumPy v1.11 Manual (original) (raw)

numpy.random.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 > 0 as of Numpy 1.10, should be > 1 for earlier versions. nonc : float Non-centrality, should be non-negative. size : int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), thenm * n * k samples are drawn. Default is None, in which case a single value is returned.

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

[R245]	Delhi, M.S. Holla, “On a noncentral chi-square distribution in the analysis of weapon systems effectiveness”, Metrika, Volume 15, Number 1 / December, 1970.

[R246]	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-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-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-noncentral_chisquare-1_02_00.png