Continuous distributions

Chi-square distribution

  • Story. If \(X_1\), \(X_2\), …, \(X_n\) are Gaussian distributed, \(X_1^2 + X_2^2 + \cdots + X_n^2\) is \(χ^2\) -distributed. See also the story of the Gamma distribution, below.
  • Example. The sample variance of \(ν−1\) independent and identically distributed Gaussian random variables, after scaling, is Chi-square distributed. This is the most common use case of the Chi-square distribution.
  • Parameters. There is only one parameter, the degrees of freedom \(ν\).
  • Support. The Chi-square distribution is supported on the positive real numbers.
  • Probability density function.
    \(\begin{align}
    f(y;\nu) \equiv \chi^2_n(x;\nu) = \frac{1}{2^{\nu/2}\,\Gamma\left(\frac{\nu}{2}\right)}\,
    x^{\frac{\nu}{2}-1}\,\mathrm{e}^{-y/2}
    \end{align}\)
  • Usage

    • Package Syntax
    NumPy np.random.chisquare(nu)
    SciPy scipy.stats.chi2(nu)
    Stan chi_square(nu)

  • Related distributions. The Chi-square distribution is a special case of the Gamma distribution with \(α=ν/2\) and \(β=1/2\).
params = [dict(name='ν', start=1, end=20, value=10, step=0.01)]
app = distribution_plot_app(x_min=0,
                            x_max=40,
                            scipy_dist=st.chi2,
                            params=params, 
                            x_axis_label='y',
                            title='Chi-square')
bokeh.io.show(app, notebook_url=notebook_url)