# Lecture 7. A) Random Sample

Let $X=\left(X_{1}..X_{n}\right)$ be an n-dimensional random vector. The random variables $X_{1}..X_{n}$ constitute a random sample if they are (mutually) independent and have identical (marginal) distributions.
It follows that if $X$ is a random sample from distribution $F\left(\cdot\right)$, then
$F_{X_{1}..X_{n}}\left(x_{1}..x_{n}\right)\underset{(independence)}{\underbrace{=}}\Pi_{i=1}^{n}F_{X_{i}}\left(x_{i}\right)\underset{(F_{X_{i}}=F,\,\forall i)}{\underbrace{=}}\Pi_{i=1}^{n}F\left(x_{i}\right)$.