Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ t-distribution ▷ Definition

Definition: Let $Z$ and $V$ be independent random variables following a standard normal distribution and a chi-squared distribution with $\nu$ degrees of freedom, respectively:

\[\label{eq:snorm-chi2} \begin{split} Z &\sim \mathcal{N}(0,1) \\ V &\sim \chi^{2}(\nu) \; . \end{split}\]

Then, the ratio of $Z$ to the square root of $V$, divided by the respective degrees of freedom, is said to be $t$-distributed with degrees of freedom $\nu$:

\[\label{eq:t} Y = \frac{Z}{\sqrt{V/\nu}} \sim t(\nu) \; .\]

The $t$-distribution is also called “Student’s $t$-distribution”, after William S. Gosset a.k.a. “Student”.


Metadata: ID: D147 | shortcut: t | author: JoramSoch | date: 2021-04-21, 07:53.