11.2 The $t$ Distribution

Parameters: $𝜽=\nu$ with $\nu >0$ called the degrees of freedom.

1. $f_X(x;𝜽)=\frac{1}{\sqrt{\nu }B(\frac{\nu }{2},\frac{1}{2}){(1+\frac{x^2}{\nu })}^{\frac{\nu +1}{2}}}$ for ,

2. $𝖤\left[X\right]=0$ when $\nu >1$ (otherwise not defined),

3. $\mathrm{𝖵𝖺𝗋}\left[X\right]=\frac{\nu }{\nu -2}$ when $\nu >2$.

We write $X\sim t_{\nu }$.

Transformations: If $Z$ and $V$ are independent with $Z\sim N(0,1)$-distributed and $V$ $\sim {\chi }_{\nu }^2$, then

has a $t_{\nu }$ distribution.

Usage: Used in statistics as the distribution of the test statistic for the test of a hypothesis about the mean in a normal sample with unknown variance, a so-called $t$ test.

Special: the $t_1$ distribution is the Cauchy distribution, which has density

Other: The $t$ distribution looks like a normal distribution but has heavier tails. For $\nu$ going to infinity the $t$ distribution converges to a normal distribution. Ignoring the normalisation constant, the density is

As $\nu \to \mathrm{\infty }$ the first term in the product $\to 1$. Since ${(1+y/v)}^v\to e^x$ as $v\to \mathrm{\infty }$, as $\nu \to \mathrm{\infty }$, the second term in the product $\to \exp (-x^2/2)$.