\begin{definition}[Definition:Random Sample]
Let $X_i$ be a random variable with $\Img {X_i} = \Omega$, for all $1 \le i \le n$.
Let $F_i$ be the cumulative distribution function of $X_i$ for all $1 \le i \le n$. 
We say that $X_1, X_2, \ldots, X_n$ form a '''random sample''' of size $n$ if: 
:$X_i$ and $X_j$ are independent if $i \ne j$
:$\map {F_1} x = \map {F_i} x$ for all $x \in \Omega$
for all $1 \le i, j \le n$.
If $X_1, X_2, \ldots, X_n$ form a '''random sample''', they are said to be '''independent and identically distributed''', commonly abbreviated '''i.i.d'''.
\end{definition}