Probability Distributions and their Stories

• 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)