Student's t distribution

The Student's $t$ distribution is a real, continuous Probability distribution commonly used in statistical inference. Given a standard-normal-distributed Random variable $Z\sim \mathcal{N}(0,1)$ and a chi-square-distributed one $X\sim \chi ^{2}_{n}$, the Probability density function is

$Z$ may follow a more general Gaussian distribution, though that is considered to be an extension of the $t$ distribution.

The $t$ distribution differs from the more usual Gaussian because it has much heavier tails, meaning there is less of a focus on the center. As the $\chi ^{2}$ distribution, when $n$ grows large the $t$ distribution converges back to a Gaussian. $n\simeq 30$ is a good practical number where the $t$ distribution becomes a good approximation of a Gaussian.

It's most useful when the amount of available data is small or when outliers are present.

Relation to other distributions#

If $n=1$, the tails are as heavy as the distribution allows: in this case it goes back to the Cauchy distribution.