Definition:Random Sample

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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.