Definition:Random Sample
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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