Definition:Student's t-Distribution

Definition
Let $X$ be a continuous random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

$X$ is said to have a $t$-distribution if it has probability density function:


 * $\displaystyle f_X \left({x}\right) = \frac {\Gamma \left({\frac {k + 1} 2}\right)} {\sqrt {\pi k} \Gamma \left({\frac k 2}\right)} \left({1 + \frac {x^2} k}\right)^{- \frac {k + 1} 2}$

for some $t \in \R_{> 0}$.

This is written:


 * $X \sim t_k$