Student's t-distribution

Definition

Let $\nu > 0$ be a real number. Student's t-distribution with $\nu$ degrees of freedom, written $t_\nu$, is the law on $\mathbb{R}$ with the density $$f(t) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu \pi}\, \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-(\nu + 1)/2}, \qquad t \in \mathbb{R}.$$

Remarks

Let $Z$ be standard normal and $V \sim \chi^2_\nu$ (see chi-squared distribution), independent. Then $Z / \sqrt{V / \nu} \sim t_\nu$. For an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample of size $n \ge 2$, $\frac{\bar{X}_n - \mu}{S_n / \sqrt{n}} \sim t_{n - 1}$, with the sample mean $\bar{X}_n$ and the sample standard deviation $S_n$. The mean is $0$ for $\nu > 1$, and the variance is $\nu / (\nu - 2)$ for $\nu > 2$. For $\nu = 1$, $t_1$ is the standard Cauchy distribution. As $\nu \to \infty$, the density of $t_\nu$ converges to the standard normal density.

Explanations

L1 human

The t-distribution looks like the bell curve, but it gives more probability to values far from $0$. It takes into account the extra uncertainty when the standard deviation comes from a small sample. With many degrees of freedom, it is almost the normal law.