Definition:Random Sample

Definition

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.

Sources

• 1974: H.T. Hayslett, MS: Statistics Made Simple (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$: Pictorial Description of Data: Introduction
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): random sample
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): random
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): random
• 2011: Morris H. DeGroot and Mark J. Schervish: Probability and Statistics (4th ed.): $3.7$: Multivariate Distributions: Definition $3.7.6$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): random sample