Procedures used to draw conclusions about population characteristics based on the information contained in a sample drawn from this population.
Let \chi_k^2 denotes a \chi^2 distribution with k degrees of freedom
Let X \sim N(0,1) and U \sim \chi_n^2 independent, then:
\frac{X}{\sqrt{\frac{U}{n}}} \sim t_n
Let X_1, \cdots, X_n \stackrel{i.i.d.}{\sim} N(\mu, \sigma). Then:
\begin{aligned} E(\bar X_n) = \mu, &&& V(\bar X_n) = \frac{\sigma^2}{n}, &&& \bar X_n \sim N(\mu, \frac{\sigma}{\sqrt{n}}) \end{aligned}
\begin{aligned} \bar X_n \stackrel{n \to \infty}{\to} \mu &\text{ in probability} \\ Var(X_n) \stackrel{n \to \infty}{\to} 0 & \\ \end{aligned}
Let X_1, \cdots, X_n be i.i.d. and E(X_i) = \mu and V(X_i) = \sigma^2 < \infty. Then:
\frac{\bar X_n - \mu}{\frac{\sigma}{\sqrt n}} \stackrel{n \to \infty}{\to} N(0, 1) \text{ in distribution}
F_n(x) = \frac{1}{n} \sum_{i=1}^n 1_{(-\infty, x]} (X_i)
Where 1_A(x) is the indicator function equal to one if x \in A and zero otherwise.
As n \to \infty,
\sup_{t \in \mathbf{R}} |F_n(t) - F(t)| \to 0
with probability 1.