When we have $n$ observations, the Student's t-distribution has $k=(n-1)$ degrees of freedom. The [[Random Variable]] follow the form: $ t_{n-1} \sim \frac{Z}{\sqrt {(V/k)}}, \quad \text{where} \begin{cases} Z \sim \mathcal N(0,1)\\[2pt] V \sim \chi_k^2 \\[2pt] Z \perp V \end{cases} $ **Assumptions:** - $Z$: A standard [[Gaussian Distribution|Gaussian]] r.v. $Z \sim \mathcal N(0,1)$ that represents the deviation of a test statistic (e.g. sample mean difference). - $V$: A [[Chi-Square Distribution|Chi-Square]] r.v. $V \sim \chi_k^2$ with $k$ degrees of freedom. It represents the number of independent pieces used to estimate variability. - $Z \perp V$: According to [[T-Test#Cochran’s Theorem|Cochran’s Theorem]] the sample variance $S_n^2$ is independent of the sample mean $\bar X_n$, denoted as $S_n^2 \perp \bar X_n$. ![[student-t-distribution.png|center|400]] When $k \to \infty$ we see that $V \over k$converges to $\mathbb E[Z_1^2]$. For a standard Gaussian, this is equal to its variance $(=1)$. Thus we can concluded that we large $k$, the more will the t-distribution look like a $\mathcal N(0,1)$. $ \frac{V}{k} = \frac{Z_1^2+\dots+Z_k^2}{k} \xrightarrow[n\to \infty]{\mathbf P}\mathbb E[Z_1^2]=1 $