Chi

The Chi random variable with $N$ degrees of freedom is

$$t = \sum_{i=1}^N \; \vert x_i\vert,$$

The mean of $t$ is

$$\mu = {\cal E}\{t\}=\sqrt{2/\pi} N\sigma,$$

and the variance of $t$ is

$${\cal E}\{(t-\mu)^2\}=N\sigma^2-\mu^2.$$

If $N=1$,

$$\log p(t)= \frac{1}{2} \log\left({2\over \pi} \right) - \frac{t^2}{2}.$$