Expectation of Student's t-Distribution

Theorem
Let $k$ be a strictly positive integer.

Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom.

Then the expectation of $X$ is equal to $0$ for $k > 1$, and does not exist otherwise.

Proof
From the definition of the Student's t-Distribution, $X$ has probability density function:


 * $\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}$

with $k$ degrees of freedom for some $k \in \R_{>0}$.

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


 * $\ds \expect X = \int_{-\infty}^\infty x \map {f_X} x \rd x$

So for $k > 1$:

When $k = 1$, we have the natural log evaluated at infinity where it is undefined.

Hence:

Therefore, the expectation of $X$ does not exist: