Example: $\chi ^2$ distribution

Definition 3 ($\chi ^2$ Distribution)   The function

$$\fbox{$ \displaystyle \rho(x) \equiv \frac{1}{2^{n/2} \Gamma(n/2)}x^{n/2-1} e^{-x/2} \theta(x) $}$$ (29)

is the probability density of the $\chi ^2$ distribution with $n$ degrees of freedom. Here,

$$\Gamma(z) \equiv \int_{0}^{\infty}dx x^{z-1} e^{-x}$$ (30)

is the Gamma function and

$$\theta(x)= \left \{ \begin{array}[h]{c} 0, x < 0 \\ 1, x \ge 0 \end{array}\right.$$ (31)

is the unit-step (Heavyside) function.