3.6.1 The t-distribution

<< 3.6 t- and F-Distributions | 3.6.2 The F-distribution >>

Definition: t-distribution

$X\sim t(\nu )$

Student’s t-distribution with $\nu$ degrees of freedom. Used when population variance is unknown and estimated from sample data.

• pdf: $\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)\sqrt{\nu \pi }}{\left(1+\frac{x^2}{\nu }\right)}^{-\frac{\nu +1}{2}}$, $x\in \mathbb{R}$, $\nu >0$
• mean: $0$ (for $\nu >1$)
• var: $\frac{\nu }{\nu -2}$ (for $\nu >2$)
• mgf: Does not exist

Key properties:

• Symmetric around 0, bell-shaped like normal distribution
• Heavier tails than standard normal
• As $\nu \to \infty$, $t(\nu )\to N(0,1)$
• For $\nu =1$, becomes Cauchy distribution

Construction: If $U\sim N(0,1)$ and $V\sim {\chi }^2(\nu )$ independently, then $T=\frac{U}{\sqrt{V/\nu }}\sim t(\nu )$

Link to original

Proof

Let

• $W\sim N(0,1)$
• $V\sim {\chi }^2(r)$
• $W$ and $V$ independent

Then joint pdf of $W$ and $V$ is the product of the pdf of $W$ and that of $V$:

$$h(w,v)=\{\begin{array}{ll}\frac{1}{2\pi }{e}^{-w^2/2}\frac{1}{\Gamma (r/2)2^{r/2}}{v}^{r/2-1}{e}^{-v/2}& -\infty <w<\infty ,0<v<\infty \\ 0& \text{elsewhere}\end{array}$$

Let random variable $T$ defined by $T=\frac{V}{\sqrt{V/r}}$

TODO