Definition:Random Sample (Probability Theory)
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 $1 \le i, j \le n$.
Continuous Distribution
Let $X$ be a continuous random variable with a known probability distribution.
Let $\map f X$ be the frequency function of $X$.
A random sample of values of $X$ is obtained by selecting values $s$ such that:
- $\map \Pr {s \in \openint x {x + \delta x} } = \map f x \delta x$
That is, the probability that $s$ is in a given interval is proportional to the length of the interval.
Discrete Distribution
Let $X$ be a discrete random variable with a known probability distribution.
Let $\map f X$ be the frequency function of $X$.
A random sample of values of $X$ is obtained by selecting values $s$ such that:
- $\map \Pr {s = x_i} = \map f {x_i}$
That is, the probability that $s$ is a given value is determined completely by the frequency function of $X$ at that value.
Methods of Generation
Tables of random samples from certain probability distributions are available commercially.
However, it is usual nowadays for simulations or Monte Carlo models to use a computer to generate effectively random samples.
Also known as
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.
Also see
- Results about random samples in the context of probability theory can be found here.
Sources
- 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 $3.7.6$