Expectation of Chi Distribution

Theorem
Let $n$ be a strictly positive integer.

Let $X \sim \chi_n$ where $\chi_n$ is the chi distribution with $n$ degrees of freedom.

Then the expectation of $X$ is given by:


 * $\expect X = \sqrt 2 \dfrac {\map \Gamma {\paren {n + 1} / 2} } {\map \Gamma {n / 2} }$

where $\Gamma$ is the gamma function.

Proof
From the definition of the chi distribution, $X$ has probability density function:


 * $\displaystyle \map {f_X} x = \dfrac 1 {2^{\paren {n / 2} - 1} \map \Gamma {n / 2} } x^{n - 1} e^{- x^2 / 2}$

From the definition of the expected value of a continuous random variable:


 * $\displaystyle \expect X = \int_0^\infty x \map {f_X} x \rd x$

So: