Student's t Distribution

Compact study note.

Summary

Student's t distribution is a continuous distribution with heavier tails than the standard normal. Its single parameter is degrees of freedom.^[1]

If $Z \sim N (0, 1)$ , $V \sim χ_{ν}^{2}$ , and $Z$ is independent of $V$ , then $T = Z / \sqrt{V / ν} \sim t_{ν}$ .

$ν > 0$ controls tail thickness. Larger $ν$ makes the distribution closer to $N (0, 1)$ .

$ν > 0$ .

$R$ .

$f_{T} (t) = \frac{Γ ((ν + 1) / 2)}{\sqrt{ν π} Γ (ν / 2)} {(1 + t^{2} / ν)}^{- (ν + 1) / 2}$ .

No elementary closed-form CDF; it is evaluated numerically.

$E [T] = 0$ for $ν > 1$ ; $Var (T) = ν / (ν - 2)$ for $ν > 2$ . The ordinary MGF does not exist for nonzero $t$ .

The one-sample t statistic has one $t_{n - 1}$ distribution under normal sampling with unknown variance.

With $n = 10$ normal observations and unknown variance, the usual one-sample statistic uses $ν = 9$ degrees of freedom.

Defining the distribution only through test-statistic usage.
Listing two parameters when standard Student's t has one degree-of-freedom parameter.

NIST/SEMATECH, e-Handbook of Statistical Methods, "1.3.6.6 Gallery of Distributions", https://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm ↩︎