Student's t-distribution
Notation:
$X \sim t_{(\nu)}$
Probability Density Function:
$f(x)=\frac{\Gamma \left( \frac{\nu+1}{2} \right)
}{\sqrt{\nu\pi}\ \Gamma \left( \frac{\nu}{2} \right)} \left( 1+\frac{x^2}{\nu}
\right)^{-\frac{\nu+1}{2}}$
where:
- $\nu > 0$
- $x \in \mathbb{R} $
Mean:
$\mathrm{E}[X]=0$ for $\nu>1$
Variance:
$\mathrm{Var}[X]=\sigma^2=\dfrac{\nu}{\nu-2}$ for $\nu>2$
References:
-
Student's t-distribution - Wikipedia
-
Student's t-distribution - Wolfram MathWorld